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R0019-10296 · q-20261003-r06-function-graph-0096
The declared domain is D={30, 31} and codomain is B={10, 11, 12}. Is the ordered-pair set {(30, 10), (31, 11)} exactly the graph of a total function D→B?
G001 — The declared domain is D={30, 31} and codomain is B={10, 11, 12}. Is the ordered-pair set {(30, 10), (31, 11)} exactly the graph of a total function D→B?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
raw domain=[30, 31]
After
sorted distinct declared domain=[30, 31]
Conditions: none
The domain is the declared finite integer subset,not automatically all R.The input list must already be distinct.The first coordinate is input,the second output.
Rule: function_graph_declared_domain_scope · Uses: G001
Before
codomain=[10, 11, 12];raw pair rows=[[30, 10], [31, 11]]
After
ordered-pair set=[[30, 10], [31, 11]]
Conditions: none
This protocol treats the graph as a set:deduplicate identical pairs,but retain different y values at one x.This differs from a raw-row protocol requiring one original row per input.
Rule: graph_set_duplicate_pairs_count_once · Uses: G001, S001
Before
pair=[30, 10];domain=[30, 31];codomain=[10, 11, 12]
After
both coordinates typed=Yes
Conditions: none
Every pair must lie in D×B.Do not ignore outside-domain or outside-codomain pairs after finding legal points for some inputs.
Rule: graph_every_pair_type_guard · Uses: G001, S002
Before
pair=[31, 11];domain=[30, 31];codomain=[10, 11, 12]
After
both coordinates typed=Yes
Conditions: none
Every pair must lie in D×B.Do not ignore outside-domain or outside-codomain pairs after finding legal points for some inputs.
Rule: graph_every_pair_type_guard · Uses: G001, S003
Before
declared input=30;normalized pair set=[[30, 10], [31, 11]]
After
distinct output fiber=[10];count=1
Conditions: none
Each declared input needs exactly one distinct output.At most one point gives single-valuedness,not totality.
Rule: graph_each_declared_input_exactly_one · Uses: G001, S004
Before
declared input=31;normalized pair set=[[30, 10], [31, 11]]
After
distinct output fiber=[11];count=1
Conditions: none
Each declared input needs exactly one distinct output.At most one point gives single-valuedness,not totality.
Rule: graph_each_declared_input_exactly_one · Uses: G001, S005
Before
all pair type flags=[Yes, Yes];source fiber sizes=[1, 1]
After
total function graph=Yes
Conditions: none
Typing and totality are joint conditions.An empty domain with an empty graph is valid,but extra outside pairs are not.The valid empty function does not require a nonempty codomain.
Rule: total_graph_type_and_totality_joint_gate · Uses: G001, S006
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.Yes
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: 3e5dc99dc3741de4494808ed257e34946f1524c535e4891b5f383f386c0f652fcandidate record SHA-256: 6a27e2f9b3c6e888a13cdba2db49ad00183df726020d1f051914f1204eed422bsemantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED