# Question 6264: All Minimum Depths in a Given MP Protocol

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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.

R0011-06264 · q-20261009-advance-20261013-r04-proof-0014



## Original question and givens

Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.


- G001 — Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.





## 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-proof](https://ocme.evemisslab.com/en/mko/mko-proof/)





## Step-by-step derivation

S001 · impliesBefore

n=8;given facts=[0, 1, 3];original indexed rules=[[0, 0, 1], [1, 1, 2], [2, 2, 3], [3, 3, 4], [4, 4, 5], [5, 5, 6], [6, 6, 7]]

After

use only given facts and single-premise MP;not-derived does not mean False

Conditions: none

Atoms use0-based indices and a rule i→j requires an already derived i.A cycle without facts cannot generate conclusions.Not-derived is not False,and this finite check is not general logical completeness.

Step identifiers and dependenciesRule: given_unary_MP_scope · Uses: G001





S002 · impliesBefore

phase=base;given facts=[0, 1, 3]

After

phase=base;active_rules=[0,1,2,3,4,5,6]

Conditions: none

Start from all given facts,retaining original rule-row indices.Atoms with no derived premises remain not-derived.

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





S003 · impliesBefore

phase=base;given fact=0

After

phase=base;derive atom=0;depth=0;basis=given

Conditions: none

A supplied fact has depth0 and needs no MP antecedent;other atoms are not silently made facts.

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





S004 · impliesBefore

phase=base;given fact=1

After

phase=base;derive atom=1;depth=0;basis=given

Conditions: none

A supplied fact has depth0 and needs no MP antecedent;other atoms are not silently made facts.

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





S005 · impliesBefore

phase=base;given fact=3

After

phase=base;derive atom=3;depth=0;basis=given

Conditions: none

A supplied fact has depth0 and needs no MP antecedent;other atoms are not silently made facts.

Step identifiers and dependenciesRule: horn_given_fact · Uses: G001, S004





S006 · calculationBefore

base MP depth for atom2 := sum(0, 1)

After

base MP depth for atom2 = 1

Conditions: none

One MP application adds1 to the already known antecedent depth.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

phase=base;rule[1]=1→2;antecedent depth=0

After

phase=base;derive atom=2;depth=1;from=1;rule=1

Conditions: none

An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order.

Step identifiers and dependenciesRule: horn_MP_derivation · Uses: G001, S006





S008 · calculationBefore

base MP depth for atom4 := sum(0, 1)

After

base MP depth for atom4 = 1

Conditions: none

One MP application adds1 to the already known antecedent depth.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · impliesBefore

phase=base;rule[3]=3→4;antecedent depth=0

After

phase=base;derive atom=4;depth=1;from=3;rule=3

Conditions: none

An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order.

Step identifiers and dependenciesRule: horn_MP_derivation · Uses: G001, S008





S010 · calculationBefore

base MP depth for atom5 := sum(1, 1)

After

base MP depth for atom5 = 2

Conditions: none

One MP application adds1 to the already known antecedent depth.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · impliesBefore

phase=base;rule[4]=4→5;antecedent depth=1

After

phase=base;derive atom=5;depth=2;from=4;rule=4

Conditions: none

An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order.

Step identifiers and dependenciesRule: horn_MP_derivation · Uses: G001, S010





S012 · calculationBefore

base MP depth for atom6 := sum(2, 1)

After

base MP depth for atom6 = 3

Conditions: none

One MP application adds1 to the already known antecedent depth.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · impliesBefore

phase=base;rule[5]=5→6;antecedent depth=2

After

phase=base;derive atom=6;depth=3;from=5;rule=5

Conditions: none

An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order.

Step identifiers and dependenciesRule: horn_MP_derivation · Uses: G001, S012





S014 · calculationBefore

base MP depth for atom7 := sum(3, 1)

After

base MP depth for atom7 = 4

Conditions: none

One MP application adds1 to the already known antecedent depth.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · impliesBefore

phase=base;rule[6]=6→7;antecedent depth=3

After

phase=base;derive atom=7;depth=4;from=6;rule=6

Conditions: none

An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order.

Step identifiers and dependenciesRule: horn_MP_derivation · Uses: G001, S014





S016 · impliesBefore

phase=base;edge closure/depth audits=[[0, 0, 1, 0, 0, Yes], [1, 1, 2, 0, 1, Yes], [2, 2, 3, 1, 0, Yes], [3, 3, 4, 0, 1, Yes], [4, 4, 5, 1, 2, Yes], [5, 5, 6, 2, 3, Yes], [6, 6, 7, 3, 4, Yes]]

After

phase=base;depths=[0,0,1,0,1,2,3,4]

Conditions: none

Every derived atom has a displayed chain of that length.The reached set is closed under every given edge and satisfies the depth bounds,so any shorter chain would force a smaller label.These labels are shortest;null means not-derived in this rule model,not False.

Step identifiers and dependenciesRule: horn_closed_depth_certificate · Uses: G001, S015









## Reverse checks


- S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Recompute base MP depth for atom7 with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S012 · premise_and_result_audit — Recompute base MP depth for atom6 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute base MP depth for atom5 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute base MP depth for atom4 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute base MP depth for atom2 with exact rational arithmetic, not rounded decimals.
- 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.





## Result

[0, 0, 1, 0, 1, 2, 3, 4]

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: 9a10371673a93556332fb0857a405a96c871ab07d1079d36346c5b3847c3aad6
- candidate record SHA-256: ffefb1c9bfbbf120bb86741da472794f7d5ad77bbebd9b69a95578c43459eefc
- semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80
- question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r04-proof-0014.json)