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R0070-35548 · q-20260913-finite-function-mapping-0003
Let X={x3-1, x3-2, x3-3, x3-4} and Y={y3-1, y3-2, y3-3, y3-4, y3-5}. The finite correspondence is x3-2↦y3-1, x3-3↦y3-1, x3-4↦y3-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
G001 — Let X={x3-1, x3-2, x3-3, x3-4} and Y={y3-1, y3-2, y3-3, y3-4, y3-5}. The finite correspondence is x3-2↦y3-1, x3-3↦y3-1, x3-4↦y3-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [qf-finite-function-mapping, [codomain, domain, pairs, scenario]]
After
derived = [function, null, null]
Conditions: none
Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules.
Rule: R70_SOURCE_bank_specific_interface · Uses: G001
Before
source roles = [[x3-1, x3-2, x3-3, x3-4], [y3-1, y3-2, y3-3, y3-4, y3-5], [[x3-2, y3-1], [x3-3, y3-1], [x3-4, y3-1]]]
After
derived = [one RAW assignment per domain input]
Conditions: none
THIS original finite-function bank requires exactly ONE RAW assignment row for every input.Do not borrow another bank’s duplicate-pair normalization.The codomain may exceed the range;check multiplicity,missing inputs and outside values directly.
Rule: Function_ORIGINAL_labels_RAW_rows_NO_legacy_dedup · Uses: G001, S001
Before
source roles = [0, x3-2, y3-1]
After
derived = [Yes]
Conditions: none
Check EACH RAW row for x∈A and y∈B.Retain duplicate rows and row indices;never discard a bad row and then announce a function.
Rule: Function_EACH_RAW_assignment_domain_and_codomain · Uses: G001, S002
Before
source roles = [1, x3-3, y3-1]
After
derived = [Yes]
Conditions: none
Check EACH RAW row for x∈A and y∈B.Retain duplicate rows and row indices;never discard a bad row and then announce a function.
Rule: Function_EACH_RAW_assignment_domain_and_codomain · Uses: G001, S003
Before
source roles = [2, x3-4, y3-1]
After
derived = [Yes]
Conditions: none
Check EACH RAW row for x∈A and y∈B.Retain duplicate rows and row indices;never discard a bad row and then announce a function.
Rule: Function_EACH_RAW_assignment_domain_and_codomain · Uses: G001, S004
Before
source roles = [x3-1, []]
After
derived = [0, No]
Conditions: none
For EVERY domain input,check EXACTLY ONE RAW assignment.Missing inputs fail;even two identical-output rows cannot silently deduplicate in this ORIGINAL contract.
Rule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S005
Before
source roles = [x3-2, [y3-1]]
After
derived = [1, Yes]
Conditions: none
For EVERY domain input,check EXACTLY ONE RAW assignment.Missing inputs fail;even two identical-output rows cannot silently deduplicate in this ORIGINAL contract.
Rule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S006
Before
source roles = [x3-3, [y3-1]]
After
derived = [1, Yes]
Conditions: none
For EVERY domain input,check EXACTLY ONE RAW assignment.Missing inputs fail;even two identical-output rows cannot silently deduplicate in this ORIGINAL contract.
Rule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S007
Before
source roles = [x3-4, [y3-1]]
After
derived = [1, Yes]
Conditions: none
For EVERY domain input,check EXACTLY ONE RAW assignment.Missing inputs fail;even two identical-output rows cannot silently deduplicate in this ORIGINAL contract.
Rule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S008
Before
source roles = [[x3-1], [], [], [y3-1], [y3-1, y3-2, y3-3, y3-4, y3-5]]
After
derived = [No]
Conditions: none
A function A→B requires absence of ALL three violations.Different inputs may share an output;injectivity/surjectivity is not required.The actual range does not replace the declared B.
Rule: Function_COMPLETE_total_single_valued_into_declared_codomain · Uses: G001, S009
S010 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S009 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S008 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S007 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S006 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S005 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S004 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S003 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.No
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 707f7fad6289ffbfc05f8a06bd3339b709cf03d6894ea13807716f3a89eee52ecandidate record SHA-256: 02730d80457f12e3d7977d62b7f79e708907ce75d3c0bb74d2d1cdb6f18eda38semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dabLean: NOT_RUN · human review: NOT_PERFORMED