Original question and givens
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[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=[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.
Step-by-step derivation
S001 · implies
Before
n=8;given facts=[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 dependencies
Rule: given_unary_MP_scope · Uses: G001
S002 · implies
Before
phase=base;given facts=[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 dependencies
Rule: horn_closure_start · Uses: G001, S001
S003 · implies
Before
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 dependencies
Rule: horn_given_fact · Uses: G001, S002
S004 · implies
Before
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 dependencies
Rule: horn_given_fact · Uses: G001, S003
S005 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
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 dependencies
Rule: horn_MP_derivation · Uses: G001, S005
S007 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
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 dependencies
Rule: horn_MP_derivation · Uses: G001, S007
S009 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
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 dependencies
Rule: horn_MP_derivation · Uses: G001, S009
S011 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
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 dependencies
Rule: horn_MP_derivation · Uses: G001, S011
S013 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · implies
Before
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 dependencies
Rule: horn_MP_derivation · Uses: G001, S013
S015 · implies
Before
phase=base;edge closure/depth audits=[[0, 0, 1, null, 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=[null,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 dependencies
Rule: horn_closed_depth_certificate · Uses: G001, S014