Question 6279: Goal Depths Before and After Removing Original Rules

Original question and givens

Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[2], goal=2. Give (depth before or not-derived, depth after or not-derived, did derivability change?).

• G001 — Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[2], goal=2. 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.

• mko-proof

Step-by-step derivation

n=3;given facts=[0];original indexed rules=[[0, 0, 1], [1, 1, 2], [2, 0, 2]]

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

phase=before;given facts=[0]

phase=before;active_rules=[0,1,2]

phase=before;given fact=0

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

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

before MP depth for atom1 = 1

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

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

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

before MP depth for atom2 = 1

phase=before;rule[2]=0→2;antecedent depth=0

phase=before;derive atom=2;depth=1;from=0;rule=2

phase=before;edge closure/depth audits=[[0, 0, 1, 0, 1, Yes], [1, 1, 2, 1, 1, Yes], [2, 0, 2, 0, 1, Yes]]

phase=before;depths=[0,1,1]

remove original rule rows=[2]

retained original indices=[0, 1]

phase=after;given facts=[0]

phase=after;active_rules=[0,1]

phase=after;given fact=0

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

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

after MP depth for atom1 = 1

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

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

after MP depth for atom2 := sum(1, 1)

after MP depth for atom2 = 2

phase=after;rule[1]=1→2;antecedent depth=1

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

phase=after;edge closure/depth audits=[[0, 0, 1, 0, 1, Yes], [1, 1, 2, 1, 2, Yes]]

phase=after;depths=[0,1,2]

goal=2;before=1;after=2

derivability changed=No

Reverse checks

• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 after MP depth for atom2 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 after MP depth for atom1 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 — 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 — Recompute before 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 — 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, 2, No)

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.