# Question 35525: Boundary of the Basic Counting Principles

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0070-35525 · q-20260914-basic-counting-principle-0005



## Original question and givens

A task can be completed through two explicitly disjoint branches with 6 and 7 outcomes. What is the total?


- G001 — A task can be completed through two explicitly disjoint branches with 6 and 7 outcomes. What is the total?





## Definition and scope

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.


- [mko-counting-principle](https://ocme.evemisslab.com/en/mko/mko-counting-principle/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-basic-counting-principle, [branch_counts, conditional_stage_witness, mode, overlap_witness, stages]]

After

derived = [counting, 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.

Step identifiers and dependenciesRule: R70_SOURCE_bank_specific_interface · Uses: G001





S002 · impliesBefore

source roles = [6, 7, addition_disjoint]

After

derived = [6, 7]

Conditions: none

Both branch counts are GIVEN positive integers.Addition still requires the disjointness/overlap premise,not merely two numbers.

Step identifiers and dependenciesRule: Counting_TWO_GIVEN_branch_cardinalities · Uses: G001, S001





S003 · calculationBefore

Counting disjoint union cardinality := sum(6, 7)

After

Counting disjoint union cardinality = 13

Conditions: none

This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

source roles = [6, 7]

After

derived = [13]

Conditions: none

The branches are GIVEN disjoint,so every union outcome is counted once:|A∪B|=|A|+|B|.This is not a rule without the disjointness premise.

Step identifiers and dependenciesRule: Counting_DISJOINT_no_double_count · Uses: G001, S003









## Reverse checks


- 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 — Recompute Counting disjoint union cardinality with exact rational arithmetic, not rounded decimals.
- 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.





## Result

13

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 9b382b132fe57f9fd1ebe1fba066abc11632be5c881e5932a555230c54801fc8
- candidate record SHA-256: ee38977c5589f432f7bc601867bc448ac7566620590fafe0ed7f78a77c9ff55d
- semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80
- question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260914-basic-counting-principle-0005.json)