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R0010-05528 · q-20261008-advance-20261012-r07-function-mapping-0078
The complete total map f:Fin(3)→Fin(8) has images=[0, 0, 0] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. A second complete total map g on the same types has other=[0, 1, 0]. Give (increasing mismatch indices, aligned [f(i),g(i)] value pairs).
G001 — The complete total map f:Fin(3)→Fin(8) has images=[0, 0, 0] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. A second complete total map g on the same types has other=[0, 1, 0]. Give (increasing mismatch indices, aligned [f(i),g(i)] value pairs).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
f:Fin(3)→Fin(8);source indices start at0
After
complete total map,not a partial program function
Conditions: none
Every legal source element has one codomain value.Fin(0) is empty;an outside query is ill-typed,not a hole of a partial function at a legal input.
Rule: total_finite_mapping_scope · Uses: G001
Before
f input=0∈Fin(3)
After
f(0)=0∈Fin(8)
Conditions: none
Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal.
Rule: complete_mapping_entry · Uses: G001, S001
Before
f input=1∈Fin(3)
After
f(1)=0∈Fin(8)
Conditions: none
Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal.
Rule: complete_mapping_entry · Uses: G001, S002
Before
f input=2∈Fin(3)
After
f(2)=0∈Fin(8)
Conditions: none
Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal.
Rule: complete_mapping_entry · Uses: G001, S003
Before
g input=0∈Fin(3)
After
g(0)=0∈Fin(8)
Conditions: none
Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal.
Rule: complete_other_mapping_entry · Uses: G001, S004
Before
g input=1∈Fin(3)
After
g(1)=1∈Fin(8)
Conditions: none
Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal.
Rule: complete_other_mapping_entry · Uses: G001, S005
Before
g input=2∈Fin(3)
After
g(2)=0∈Fin(8)
Conditions: none
Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal.
Rule: complete_other_mapping_entry · Uses: G001, S006
Before
source index=0;f(i)=0;g(i)=0
After
different=No
Conditions: none
Compare the same source index pointwise;the value pair is[f(i),g(i)],without exchanging the maps.
Rule: pointwise_mapping_comparison · Uses: G001, S007
Before
source index=1;f(i)=0;g(i)=1
After
different=Yes
Conditions: none
Compare the same source index pointwise;the value pair is[f(i),g(i)],without exchanging the maps.
Rule: pointwise_mapping_comparison · Uses: G001, S008
Before
source index=2;f(i)=0;g(i)=0
After
different=No
Conditions: none
Compare the same source index pointwise;the value pair is[f(i),g(i)],without exchanging the maps.
Rule: pointwise_mapping_comparison · Uses: G001, S009
Before
([1], [[0, 1]])
After
functions equal on the full source=No
Conditions: none
Two complete maps of the same declared type are equal exactly when all source values agree;an empty source has no mismatches.
Rule: finite_mapping_extensionality · Uses: G001, S010
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([1], [[0, 1]])
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: d7c4211d12bbc790fb3cfc10fdf059165fb41fab6e9b642fa21f04ecdfbd2685candidate record SHA-256: a14bc4289a95812a2b89a7417081f595ab7740f1bdfcbf8786b2ca56e74323b7semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06fquestion-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fcLean: NOT_RUN · human review: NOT_PERFORMED