Original question and givens
Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 0, 0, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.
G001 — Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 0, 0, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.
Step-by-step derivation
S001 · implies
Before
source roles = [qf-mod25-measdeep-preimage-complement-law, measdeep-preimage-complement-law, Finite_preimage_law_density, []]
After
derived = [[mapping, event, target_complement, preimage, source_complement, complement_preimage, equal]]
Conditions: none
MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status.
Step identifiers and dependencies
Rule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001
S002 · implies
Before
source roles = [4, 4, [0, 0, 0, 0]]
After
derived = [[0, 0, 0, 0]]
Conditions: none
THIS source slice has carrier sizes1..7 and a typed target at EVERY original source index.Validate the WHOLE table before preimages;neither injectivity nor an inverse function is assumed.
Step identifiers and dependencies
Rule: Meas_RAW_complete_total_map_typed_NONEMPTY_carriers · Uses: G001, S001
S003 · implies
Before
source roles = [complement ORIGINAL event, 0, 0, [0, 1]]
After
derived = [Yes]
Conditions: none
Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image.
Step identifiers and dependencies
Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S002
S004 · implies
Before
source roles = [complement ORIGINAL event, 1, 0, [0, 1]]
After
derived = [Yes]
Conditions: none
Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image.
Step identifiers and dependencies
Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S003
S005 · implies
Before
source roles = [complement ORIGINAL event, 2, 0, [0, 1]]
After
derived = [Yes]
Conditions: none
Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image.
Step identifiers and dependencies
Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S004
S006 · implies
Before
source roles = [complement ORIGINAL event, 3, 0, [0, 1]]
After
derived = [Yes]
Conditions: none
Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image.
Step identifiers and dependencies
Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S005
S007 · implies
Before
source roles = [complement TARGET complement, 0, 0, [2, 3]]
After
derived = [No]
Conditions: none
Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image.
Step identifiers and dependencies
Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S006
S008 · implies
Before
source roles = [complement TARGET complement, 1, 0, [2, 3]]
After
derived = [No]
Conditions: none
Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image.
Step identifiers and dependencies
Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S007
S009 · implies
Before
source roles = [complement TARGET complement, 2, 0, [2, 3]]
After
derived = [No]
Conditions: none
Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image.
Step identifiers and dependencies
Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S008
S010 · implies
Before
source roles = [complement TARGET complement, 3, 0, [2, 3]]
After
derived = [No]
Conditions: none
Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image.
Step identifiers and dependencies
Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S009
S011 · implies
Before
source roles = [[0, 0, 0, 0], [0, 1], [2, 3]]
After
derived = [[0, 1, 2, 3], [], [], Yes]
Conditions: none
Take the TARGET complement in Fin(target_n)and the SOURCE preimage complement in Fin(source_n),then compare both constructions.This set-law family supplies no extra sigma premise;do not invent measurability or inverse-function requirements.
Step identifiers and dependencies
Rule: Meas_COMPLEMENT_BOTH_preimage_constructions_OWN_carriers_NO_sigma_requirement · Uses: G001, S010
Result
([0, 0, 0, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.