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Question 6285: Goal Depths Before and After Removing Original Rules

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0011-06285 · q-20261009-advance-20261013-r04-proof-0085

Original question and givens

Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[0], goal=1. Give (depth before or not-derived, depth after or not-derived, did derivability change?).

  • G001 — Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[0], goal=1. Give (depth before or not-derived, depth after or not-derived, did derivability change?).

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.

Step-by-step derivation

S001 · implies

Before

n=2;given facts=[0];original indexed rules=[[0, 0, 1]]

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 dependencies

Rule: given_unary_MP_scope · Uses: G001

S002 · implies

Before

phase=before;given facts=[0]

After

phase=before;active_rules=[0]

Conditions: none

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

Step identifiers and dependencies

Rule: horn_closure_start · Uses: G001, S001

S003 · implies

Before

phase=before;given fact=0

After

phase=before;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 dependencies

Rule: horn_given_fact · Uses: G001, S002

S004 · calculation

Before

before MP depth for atom1 := sum(0, 1)

After

before MP depth for atom1 = 1

Conditions: none

One MP application adds1 to the already known antecedent depth.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

phase=before;rule[0]=0→1;antecedent depth=0

After

phase=before;derive atom=1;depth=1;from=0;rule=0

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 dependencies

Rule: horn_MP_derivation · Uses: G001, S004

S006 · implies

Before

phase=before;edge closure/depth audits=[[0, 0, 1, 0, 1, Yes]]

After

phase=before;depths=[0,1]

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 dependencies

Rule: horn_closed_depth_certificate · Uses: G001, S005

S007 · implies

Before

remove original rule rows=[0]

After

retained original indices=[]

Conditions: none

Remove by original row index;duplicate-content rows still have distinct indices.Do not renumber before interpreting the removal list.

Step identifiers and dependencies

Rule: horn_original_rule_index_removal · Uses: G001, S006

S008 · implies

Before

phase=after;given facts=[0]

After

phase=after;active_rules=[]

Conditions: none

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

Step identifiers and dependencies

Rule: horn_closure_start · Uses: G001, S007

S009 · implies

Before

phase=after;given fact=0

After

phase=after;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 dependencies

Rule: horn_given_fact · Uses: G001, S008

S010 · implies

Before

phase=after;edge closure/depth audits=[]

After

phase=after;depths=[0,null]

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 dependencies

Rule: horn_closed_depth_certificate · Uses: G001, S009

S011 · implies

Before

goal=1;before=1;after=not obtained

After

derivability changed=Yes

Conditions: none

A derivability change concerns derived versus not-derived.If the shortest depth grows but the goal remains derived,the third field is still No.Removing rules cannot invent proofs.

Step identifiers and dependencies

Rule: horn_reachability_vs_depth_change · Uses: G001, S010

Reverse checks

  • 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 — Recompute before MP depth for atom1 with exact rational arithmetic, not rounded decimals.
  • 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

(1, not-derived, Yes)

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: ea004b889c4a7487b2161fc6e6855cc69c7a161478c755212dc45cccbab3b1e6
  • candidate record SHA-256: 427556ad28aa3cb530c34e375a025ab107acee7d1167e6233f7b7a3e9ef32f3c
  • semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80
  • question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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