Feature flag rule: (beta_user OR internal) AND flag_on AND NOT kill_switch
A feature flag rule, (beta_user OR internal) AND flag_on AND NOT kill_switch: sixteen rows with three enabling ones, a two-term DNF, two Karnaugh groups, and a NOR form of 8 gates against 23 NAND.
The feature is enabled for beta users or internal staff, but only while the flag is on and the kill switch is off. The engine's reading lists the three requirements: at least one of beta_user or internal, flag_on true, kill_switch false. With four inputs the table has sixteen rows, and the output is true on three of them, Σm(6, 10, 14): an internal user, a beta user, or both, each with the flag on and the switch off.
The checks: not a tautology, counterexample all four inputs FALSE; satisfiable, witness beta_user = FALSE, internal = TRUE, flag_on = TRUE, kill_switch = FALSE. The simplifier found no law that shortens the four-operator rule, and the CNF converter returns it unchanged as three clauses. The DNF converter applies distribution twice and gives (beta_user AND flag_on AND NOT kill_switch) OR (internal AND flag_on AND NOT kill_switch), two terms and six literals, which is also what the Karnaugh map derives from its two groups, on cells 10 and 14 and on cells 6 and 14, two prime implicants. Gate counts: 23 NAND, 8 NOR.
Flag rules are evaluated on every request, usually by a library that takes a list of conditions, and the most frequent bug is an audience OR that swallows the safety checks, as on the entry-rule slip page. The thing to watch here is the kill switch: it must override everything, which in the formula means it is ANDed at the top level and not placed inside the audience group. The three true rows all have kill_switch = FALSE, and that is the property to assert in a test.
enabled := (beta_user OR internal) AND flag_on AND NOT kill_switch · output
Exact checks
Truth table (16 rows)
| # | beta_user | internal | flag_on | kill_switch | enabled |
|---|---|---|---|---|---|
| 0 | FALSE | FALSE | FALSE | FALSE | FALSE |
| 1 | FALSE | FALSE | FALSE | TRUE | FALSE |
| 2 | FALSE | FALSE | TRUE | FALSE | FALSE |
| 3 | FALSE | FALSE | TRUE | TRUE | FALSE |
| 4 | FALSE | TRUE | FALSE | FALSE | FALSE |
| 5 | FALSE | TRUE | FALSE | TRUE | FALSE |
| 6 | FALSE | TRUE | TRUE | FALSE | TRUE |
| 7 | FALSE | TRUE | TRUE | TRUE | FALSE |
| 8 | TRUE | FALSE | FALSE | FALSE | FALSE |
| 9 | TRUE | FALSE | FALSE | TRUE | FALSE |
| 10 | TRUE | FALSE | TRUE | FALSE | TRUE |
| 11 | TRUE | FALSE | TRUE | TRUE | FALSE |
| 12 | TRUE | TRUE | FALSE | FALSE | FALSE |
| 13 | TRUE | TRUE | FALSE | TRUE | FALSE |
| 14 | TRUE | TRUE | TRUE | FALSE | TRUE |
| 15 | TRUE | TRUE | TRUE | TRUE | FALSE |
Plain-language reading
Beginnerenabled is true exactly when all of the following hold:
1. at least one of beta_user or internal is true (both may be true)
2. flag_on is true
3. kill_switch is false
In every other case, enabled is false.
About the operators:
- AND is true only when all of its parts are true.
- OR is inclusive: it is true when at least one part is true, including when several are.
- NOT reverses a value: true becomes false and false becomes true.
Intermediateenabled is true exactly when all of the following hold:
1. at least one of beta_user or internal is true (both may be true)
2. flag_on is true
3. kill_switch is false
In every other case, enabled is false.
Advancedenabled is true exactly when all of the following hold [conjunction ∧]:
1. at least one of beta_user or internal is true (both may be true) [inclusive disjunction ∨]
2. flag_on is true
3. kill_switch is false [negation ¬]
In every other case, enabled is false.
Formal notation: (beta_user ∨ internal) ∧ flag_on ∧ ¬kill_switch
Logic diagram
Simplification, step by step
The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.
(beta_user OR internal) AND flag_on AND NOT kill_switch
Karnaugh map
- cells m10, m14
beta_user AND flag_on AND NOT kill_switch - cells m6, m14
internal AND flag_on AND NOT kill_switch
Minterms and maxterms
enabled(beta_user, internal, flag_on, kill_switch) = Σm(6, 10, 14)enabled(beta_user, internal, flag_on, kill_switch) = ΠM(0, 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15)Variable order: beta_user, internal, flag_on, kill_switch · 16 rows
Normal forms
Conjunctive normal form (CNF)
(beta_user OR internal) AND flag_on AND NOT kill_switch3 clauses, 4 literals · complete
Disjunctive normal form (DNF)
(beta_user AND flag_on AND NOT kill_switch) OR (internal AND flag_on AND NOT kill_switch)2 terms, 6 literals · complete · Distribution ×2
NAND-only and NOR-only
NAND-only form
NAND(NAND(NAND(NAND(NAND(NAND(beta_user, beta_user), NAND(internal, internal)), flag_on), NAND(NAND(NAND(beta_user, beta_user), NAND(internal, internal)), flag_on)), NAND(kill_switch, kill_switch)), NAND(NAND(NAND(NAND(NAND(beta_user, beta_user), NAND(internal, internal)), flag_on), NAND(NAND(NAND(beta_user, beta_user), NAND(internal, internal)), flag_on)), NAND(kill_switch, kill_switch)))23 gates · verified equivalent on every row
NOR-only form
NOR(NOR(NOR(NOR(beta_user, internal), NOR(flag_on, flag_on)), NOR(NOR(beta_user, internal), NOR(flag_on, flag_on))), kill_switch)8 gates · verified equivalent on every row
The same rule as code
Python
# -*- coding: utf-8 -*-
# Generated by Tangibool (tangibool.python/1). Do not edit by hand; regenerate from the source revision.
# Model: not stated
# Source revision: reference
# Interpretation status: not stated
"""Tangibool evaluator (decision model).
Generated deterministically from the validated Boolean model; it evaluates exactly that
model. It does not establish that the model is a complete, legal, or safe statement of a
real-world rule, and evaluating examples does not verify the rule.
Source revision: reference
Interpretation status: not stated
Exporter: 1.0.0 (format tangibool.python/1; engine 1.0.0; semantics tangibool-semantics/1)
Logic digest: e38bc7c67f8a857345783c8ad32faa5a9e83e9828b516a5d2f543e988edde5e1
Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78
Inputs (dict keys, declared order) and generated parameter names:
'beta_user' -> input_beta_user
'internal' -> input_internal
'flag_on' -> input_flag_on
'kill_switch' -> input_kill_switch
Outputs (declared order) and their functions:
'enabled' -> output_enabled(inputs)
Assumptions (the declared scope, checked by in_scope(inputs)):
none; every complete assignment is in scope.
Input contract:
Every required input must be exactly True or False (type(value) is bool). Missing
inputs, None, strings, integers (including 0 and 1), and keys that are not declared
inputs raise InputError before anything is evaluated, including inputs that a
short-circuiting operator would never read. Results are always bool.
Scope:
Decision functions raise OutOfScopeError when the inputs violate an assumption; an
out-of-scope assignment is never turned into a False result. Pass check_scope=False
to evaluate without the scope check, and use in_scope(inputs) to test it separately.
Usage:
import tangibool_model as model
model.evaluate({'beta_user': False, 'internal': False, 'flag_on': False, 'kill_switch': False})
# -> {'enabled': False}
model.output_enabled({'beta_user': False, 'internal': False, 'flag_on': False, 'kill_switch': False})
# -> False
model.evaluate({'beta_user': 1, 'internal': False, 'flag_on': False, 'kill_switch': False})
# raises InputError: 1 is an int, not a bool
"""
import collections.abc as _abc
from typing import Dict, Mapping, Tuple
__all__ = [
"MODEL_INFO",
"IDENTIFIER_MAP",
"INPUT_NAMES",
"OUTPUT_NAMES",
"ASSUMPTIONS",
"InputError",
"OutOfScopeError",
"minterm_index",
"in_scope",
"evaluate",
"output_enabled",
]
MODEL_INFO = {
"format": "tangibool.python/1",
"mode": "decision",
"model": None,
"revision": "reference",
"interpretation": None,
"exporter": "1.0.0",
"engine_version": "1.0.0",
"semantics_version": "tangibool-semantics/1",
"logic_digest": "e38bc7c67f8a857345783c8ad32faa5a9e83e9828b516a5d2f543e988edde5e1",
"assumptions_digest": "d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78",
"primary_output": "enabled",
"unused_inputs": (),
"subcircuits": (),
}
# Stable ID -> generated identifier (reversible mapping; names are exact, escaped strings).
IDENTIFIER_MAP = {
"v_b5969b9da187": {"kind": "input", "name": "beta_user", "identifier": "input_beta_user"},
"v_8378caade4d3": {"kind": "input", "name": "internal", "identifier": "input_internal"},
"v_087e340e1d25": {"kind": "input", "name": "flag_on", "identifier": "input_flag_on"},
"v_fe3edc05673c": {"kind": "input", "name": "kill_switch", "identifier": "input_kill_switch"},
"f_33a09602b286": {"kind": "output", "name": "enabled", "identifier": "output_enabled"},
}
INPUT_NAMES = ("beta_user", "internal", "flag_on", "kill_switch")
OUTPUT_NAMES = ("enabled",)
ASSUMPTIONS = ()
_DECLARED_INPUTS = frozenset(INPUT_NAMES)
_SCOPE_INPUTS = ()
_ASSUMPTION_FORMULAS = {entry["name"]: entry["formula"] for entry in ASSUMPTIONS}
class InputError(ValueError):
"""Raised when inputs are missing, undeclared, or not exactly True/False.
Every required input must satisfy type(value) is bool. Missing inputs, None, strings,
integers (including 0 and 1), and keys that are not declared inputs are rejected before
anything is evaluated. A missing input is never treated as False. The individual problems
are listed in the problems attribute.
"""
def __init__(self, problems):
self.problems = tuple(problems)
super().__init__("Invalid inputs: " + "; ".join(self.problems))
class OutOfScopeError(ValueError):
"""Raised when the inputs violate the model's assumptions (its declared scope).
Outside its assumptions the rule has no decision: this error is never a False result.
The names of the violated assumptions are in the violated attribute.
"""
def __init__(self, violated):
self.violated = tuple(violated)
details = ", ".join(name + " (" + _ASSUMPTION_FORMULAS.get(name, "?") + ")" for name in self.violated)
super().__init__(
"Out of scope: the inputs violate "
+ ("assumption " if len(self.violated) == 1 else "assumptions ")
+ details
+ ". The rule has no decision outside its assumptions; this is not a False result."
)
def _short_repr(value: object) -> str:
text = repr(value)
return text if len(text) <= 60 else text[:57] + "..."
def _checked_inputs(inputs: Mapping[str, bool], required: Tuple[str, ...], check_scope: bool) -> Dict[str, bool]:
"""Validate inputs strictly before anything is evaluated and return a plain copy."""
if type(check_scope) is not bool:
raise TypeError("check_scope must be True or False")
if not isinstance(inputs, _abc.Mapping):
raise InputError(["inputs must be a mapping from input names to True/False, got " + type(inputs).__name__])
problems = []
for key in inputs:
if type(key) is not str or key not in _DECLARED_INPUTS:
declared = ", ".join(repr(name) for name in INPUT_NAMES) or "none"
problems.append("undeclared input " + _short_repr(key) + " (declared inputs: " + declared + ")")
needed = tuple(required) + (_SCOPE_INPUTS if check_scope else ())
for name in needed:
if name not in inputs:
problems.append("missing required input " + repr(name))
for key, value in inputs.items():
if type(key) is str and key in _DECLARED_INPUTS and type(value) is not bool:
problems.append(
"input " + repr(key) + " must be True or False (type bool), got "
+ type(value).__name__ + " " + _short_repr(value)
)
if problems:
raise InputError(list(dict.fromkeys(problems)))
return dict(inputs.items())
def _require_bool_args(function: str, pairs: Tuple[Tuple[str, object], ...]) -> None:
problems = [
"argument " + repr(name) + " of " + function + " must be True or False (type bool), got "
+ type(value).__name__ + " " + _short_repr(value)
for name, value in pairs
if type(value) is not bool
]
if problems:
raise InputError(problems)
def minterm_index(*values: bool) -> int:
"""Row index of an assignment over a declared variable order.
The first value is the most significant bit, so all-False is row 0. Used for indexed
minterm (sum) and maxterm (product) forms.
"""
index = 0
for value in values:
if type(value) is not bool:
raise InputError(["minterm_index() takes True/False values, got " + type(value).__name__ + " " + _short_repr(value)])
index = index * 2 + (1 if value else 0)
return index
def _scope_values(values: Dict[str, bool]) -> Tuple[Tuple[str, bool], ...]:
"""Value of every enabled, confirmed assumption for validated inputs, in declared order."""
return ()
def _require_in_scope(values: Dict[str, bool]) -> None:
violated = tuple(name for name, holds in _scope_values(values) if not holds)
if violated:
raise OutOfScopeError(violated)
def in_scope(inputs: Mapping[str, bool]) -> bool:
"""True when the inputs satisfy every assumption (the model's declared scope).
Requires the inputs the assumptions read; raises InputError for invalid inputs.
"""
values = _checked_inputs(inputs, (), True)
return all(holds for _, holds in _scope_values(values))
# ---- Formula bodies (inputs are validated by the public functions below) ----
def _compute_output_enabled(input_beta_user: bool, input_internal: bool, input_flag_on: bool, input_kill_switch: bool) -> bool:
"""Decision output 'enabled': (beta_user OR internal) AND flag_on AND NOT kill_switch"""
return (input_beta_user or input_internal) and input_flag_on and (not input_kill_switch)
# ---- Public API ----
def output_enabled(inputs: Mapping[str, bool], *, check_scope: bool = True) -> bool:
"""Decision output 'enabled' (stable ID f_33a09602b286).
'enabled' := (beta_user OR internal) AND flag_on AND NOT kill_switch
Required inputs: 'beta_user', 'internal', 'flag_on', 'kill_switch'.
Raises InputError for missing, undeclared, or non-bool inputs, and OutOfScopeError
when check_scope is True and the inputs violate an assumption.
"""
values = _checked_inputs(inputs, ("beta_user", "internal", "flag_on", "kill_switch"), check_scope)
if check_scope:
_require_in_scope(values)
output_enabled_value = _compute_output_enabled(values["beta_user"], values["internal"], values["flag_on"], values["kill_switch"])
return output_enabled_value
def evaluate(inputs: Mapping[str, bool], *, check_scope: bool = True) -> Dict[str, bool]:
"""Evaluate every output (declared order); each output is computed once.
Required inputs: 'beta_user', 'internal', 'flag_on', 'kill_switch'.
Raises InputError for missing, undeclared, or non-bool inputs, and OutOfScopeError
when check_scope is True and the inputs violate an assumption.
"""
values = _checked_inputs(inputs, ("beta_user", "internal", "flag_on", "kill_switch"), check_scope)
if check_scope:
_require_in_scope(values)
output_enabled_value = _compute_output_enabled(values["beta_user"], values["internal"], values["flag_on"], values["kill_switch"])
return {
"enabled": output_enabled_value,
}
PostgreSQL
-- Tangibool PostgreSQL export (tangibool.postgresql/1)
-- Target: PostgreSQL 16 or later with typed boolean inputs; tested with PostgreSQL 18.3 (PGlite 0.5.8).
-- Not a claim of compatibility with other SQL dialects.
-- Model: not stated
-- Source revision: reference
-- Interpretation status: not stated
-- Exporter: 1.0.0 (engine 1.0.0; semantics tangibool-semantics/1)
-- Logic digest: e38bc7c67f8a857345783c8ad32faa5a9e83e9828b516a5d2f543e988edde5e1
-- Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78
-- Mode: decision
--
-- Read-only: this file contains only WITH / VALUES / SELECT statements. No DDL, DML,
-- procedures, credentials, network access, or database changes. The expressions are pure
-- Boolean expressions, so their result does not depend on SQL evaluation order.
--
-- NON-NULL INPUT PRECONDITION: NULL is not a truth value of this rule. Every fragment below
-- is defined only when every input column it reads is a non-NULL boolean. The guard template
-- (section 2) marks any NULL input as 'invalid_input' and any row that violates the
-- assumptions as 'out_of_scope', and fills output columns only for 'ok' rows; a NULL in an
-- output column of another row is an envelope marker, not a third truth value. Never wrap
-- the fragments in COALESCE(..., FALSE) or otherwise choose a meaning for unknown inputs.
-- This evaluates the supplied model; it does not show that the model is a complete, legal,
-- or safe statement of a real-world rule.
--
-- Input columns (stable ID: name -> column):
-- v_b5969b9da187: beta_user -> "in_beta_user"
-- v_8378caade4d3: internal -> "in_internal"
-- v_087e340e1d25: flag_on -> "in_flag_on"
-- v_fe3edc05673c: kill_switch -> "in_kill_switch"
-- Output columns:
-- f_33a09602b286: enabled -> "out_enabled"
-- Assumptions (scope K; exported separately and never folded into an output):
-- none; every complete input row is in scope.
-- ----------------------------------------------------------------------------------------------
-- 1. Expression fragments (parenthesised; non-NULL boolean inputs required)
-- ----------------------------------------------------------------------------------------------
--
-- [out_enabled] output enabled (stable ID f_33a09602b286)
-- enabled := (beta_user OR internal) AND flag_on AND NOT kill_switch
-- PRECONDITION: "in_beta_user", "in_internal", "in_flag_on", "in_kill_switch" are non-NULL booleans.
-- (("in_beta_user" OR "in_internal") AND "in_flag_on" AND (NOT "in_kill_switch"))
--
-- ----------------------------------------------------------------------------------------------
-- 2. Guard template (read-only SELECT over an inputs list)
-- ----------------------------------------------------------------------------------------------
-- Replace the example rows between the markers with your own rows, or replace the inputs
-- CTE with a SELECT that provides the boolean columns listed in the header. status is
-- 'invalid_input' when any input is NULL, 'out_of_scope' when the assumptions are false, and
-- 'ok' otherwise; output columns are filled only for 'ok' rows (NULL elsewhere is an
-- envelope marker). Assumption columns are shown for rows with complete inputs.
WITH inputs ("row_id", "in_beta_user", "in_internal", "in_flag_on", "in_kill_switch") AS (
-- tangibool:example-rows:begin
VALUES
(1, FALSE, FALSE, FALSE, FALSE),
(2, FALSE, FALSE, FALSE, TRUE),
(3, FALSE, FALSE, TRUE, FALSE),
(4, FALSE, FALSE, TRUE, TRUE),
(5, FALSE, TRUE, FALSE, FALSE),
(6, FALSE, TRUE, FALSE, TRUE),
(7, FALSE, TRUE, TRUE, FALSE),
(8, FALSE, TRUE, TRUE, TRUE),
(9, TRUE, FALSE, FALSE, FALSE),
(10, TRUE, FALSE, FALSE, TRUE),
(11, TRUE, FALSE, TRUE, FALSE),
(12, TRUE, FALSE, TRUE, TRUE),
(13, TRUE, TRUE, FALSE, FALSE),
(14, TRUE, TRUE, FALSE, TRUE),
(15, TRUE, TRUE, TRUE, FALSE),
(16, TRUE, TRUE, TRUE, TRUE),
(17, NULL::boolean, FALSE, FALSE, FALSE)
-- tangibool:example-rows:end
),
checked AS (
SELECT
inputs.*,
CASE
WHEN "in_beta_user" IS NULL OR "in_internal" IS NULL OR "in_flag_on" IS NULL OR "in_kill_switch" IS NULL THEN 'invalid_input'
ELSE 'ok'
END AS "status"
FROM inputs
)
SELECT
"row_id",
"in_beta_user",
"in_internal",
"in_flag_on",
"in_kill_switch",
"status",
CASE WHEN "status" = 'ok' THEN (("in_beta_user" OR "in_internal") AND "in_flag_on" AND (NOT "in_kill_switch")) END AS "out_enabled"
FROM checked
ORDER BY "row_id";
SMT-LIB
; Tangibool SMT-LIB export (tangibool.smtlib/1) ; Profile: SMT-LIB 2.6 Boolean subset (QF_UF): Bool constants, define-fun, let, push/pop, assert, check-sat, echo. ; Tested with Z3 5.1.0 (z3-solver 5.2.0, WebAssembly build). ; Model: not stated ; Source revision: reference ; Interpretation status: not stated ; Exporter: 1.0.0 (engine 1.0.0; semantics tangibool-semantics/1) ; Logic digest: e38bc7c67f8a857345783c8ad32faa5a9e83e9828b516a5d2f543e988edde5e1 ; Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78 ; Mode: decision ; ; Reading results: each query block states what sat and unsat mean for its question. A sat ; answer comes with a witness or counterexample (ask for it with (get-model) after sat only); ; unsat settles that scoped query and is solver-checked, not an independently checked proof ; certificate; unknown settles nothing. Blocks are scoped with push/pop, so assertions never ; accumulate across questions. This checks the supplied model, not whether it is a complete, ; legal, or safe statement of a real-world rule. ; ; Symbols (stable ID: name -> symbol): ; v_b5969b9da187: beta_user -> in_beta_user ; v_8378caade4d3: internal -> in_internal ; v_087e340e1d25: flag_on -> in_flag_on ; v_fe3edc05673c: kill_switch -> in_kill_switch ; f_33a09602b286: enabled -> out_enabled (set-option :produce-models true) (set-info :smt-lib-version 2.6) (set-logic QF_UF) ; ---- Inputs (declared order) (declare-const in_beta_user Bool) ; beta_user (declare-const in_internal Bool) ; internal (declare-const in_flag_on Bool) ; flag_on (declare-const in_kill_switch Bool) ; kill_switch ; ---- Outputs (dependencies first) ; output enabled (stable ID f_33a09602b286): (beta_user OR internal) AND flag_on AND NOT kill_switch (define-fun out_enabled () Bool (and (or in_beta_user in_internal) in_flag_on (not in_kill_switch))) ; ---- Assumptions (the scope K; never folded into an output) ; K: conjunction of the enabled, confirmed assumptions (true when there are none) (define-fun K () Bool true) ; ---- Definitions only: nothing is asserted and no check is run. ; Add (assert ...) and (check-sat) inside (push 1) ... (pop 1) to ask a question.
LaTeX
enabled = \left(\mathit{beta\_user} + \mathit{internal}\right) \cdot \mathit{flag\_on} \cdot \overline{\mathit{kill\_switch}}The common slip
Putting the kill switch inside the audience group: (beta_user OR internal AND NOT kill_switch) AND flag_on, so that beta users bypass it. The intended rule has three true rows, all with the switch off; the slip adds rows with the switch on, and the equivalence checker would list them.
Questions
Why does the DNF repeat flag_on AND NOT kill_switch in both terms?
Distribution copies the shared conditions into each audience term, which is how a sum of products must be written. The two terms are the two Karnaugh groups, overlapping on cell 14 where both audiences apply. The factored original is shorter to read; the DNF is the list of enabling profiles.
Related
Operators and gates
- NOT gate (inverter): truth table, symbol and the NOT operator
- AND gate: truth table for 2, 3 and 4 inputs, symbol and expression
- OR gate: truth table for 2, 3 and 4 inputs, inclusive or explained
Tools
- Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns
- Boolean algebra calculator and simplifier with steps (every law named and verified)
- Karnaugh map solver: 2 to 4 variables, groups, SOP and POS, don’t-cares from assumptions
See also
- Feature flag logic: combining flags without surprises
- Entry rule slip: member OR (invited AND NOT banned) and the case that differs
- Access control rule: owner OR admin OR (editor AND NOT locked)
- Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns
- Deriving test cases from a truth table without testing every row
- Boolean logic in JavaScript: &&, ||, !, truthiness and precedence
Worked examples
Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build . Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.