Original question and givens
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.
G001 — Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.
Step-by-step derivation
S001 · implies
Before
carrier Fin(6);raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]];original vertex sequence=[3, 4, 0, 1, 0]
After
canonical undirected simple edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]];nonempty typed sequence retained
Conditions: none
Normalize undirected graph edges and reject loops/reversed duplicates.Preserve vertex-sequence order and repetitions.Repetition is not a parsing error in a candidate,though it may fail walk or simplicity qualification.
Step identifiers and dependencies
Rule: walk_simple_graph_and_ordered_candidate · Uses: G001
S002 · implies
Before
proposed step index=0;from=3;to=4;unordered step edge=[3, 4]
After
edge present=Yes
Conditions: none
Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity.
Step identifiers and dependencies
Rule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S001
S003 · implies
Before
proposed step index=1;from=4;to=0;unordered step edge=[0, 4]
After
edge present=Yes
Conditions: none
Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity.
Step identifiers and dependencies
Rule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S002
S004 · implies
Before
proposed step index=2;from=0;to=1;unordered step edge=[0, 1]
After
edge present=Yes
Conditions: none
Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity.
Step identifiers and dependencies
Rule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S003
S005 · implies
Before
proposed step index=3;from=1;to=0;unordered step edge=[0, 1]
After
edge present=Yes
Conditions: none
Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity.
Step identifiers and dependencies
Rule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S004
S006 · implies
Before
all bad proposed steps=[]
After
walk=Yes
Conditions: none
The whole step list must pass.A singleton has no proposed steps and passes vacuously as a valid length-zero walk.
Step identifiers and dependencies
Rule: complete_walk_candidate_qualification · Uses: G001, S005
S007 · calculation
Before
proposed edge length := sub(5, 1)
After
proposed edge length = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
original vertex sequence=[3, 4, 0, 1, 0]
After
sorted distinct repeated vertex labels=[0]
Conditions: none
A simple path forbids any repeated vertex,not only immediate repeats.This includes a closed walk's repeated endpoint.
Step identifiers and dependencies
Rule: simple_path_vertex_repetition_full_sequence · Uses: G001, S007
S009 · implies
Before
proposed step index=0;unordered edge=[3, 4]
After
seen at earlier step=No;distinct repeated edges so far=[]
Conditions: none
Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order.
Step identifiers and dependencies
Rule: trail_undirected_edge_first_repeat_order · Uses: G001, S008
S010 · implies
Before
proposed step index=1;unordered edge=[0, 4]
After
seen at earlier step=No;distinct repeated edges so far=[]
Conditions: none
Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order.
Step identifiers and dependencies
Rule: trail_undirected_edge_first_repeat_order · Uses: G001, S009
S011 · implies
Before
proposed step index=2;unordered edge=[0, 1]
After
seen at earlier step=No;distinct repeated edges so far=[]
Conditions: none
Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order.
Step identifiers and dependencies
Rule: trail_undirected_edge_first_repeat_order · Uses: G001, S010
S012 · implies
Before
proposed step index=3;unordered edge=[0, 1]
After
seen at earlier step=Yes;distinct repeated edges so far=[[0, 1]]
Conditions: none
Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order.
Step identifiers and dependencies
Rule: trail_undirected_edge_first_repeat_order · Uses: G001, S011
S013 · implies
Before
walk=Yes;length=4;start=3;end=0;repeated vertices=[0];repeated edges=[[0, 1]]
After
trail=No;simple path=No;closed positive-length walk=No
Conditions: none
A trail has no repeated edges,a simple path no repeated vertices,and closed additionally requires positive length and equal endpoints.These are distinct.A singleton is a trail/simple path but is not closed in this protocol;closed is not a simple-cycle certificate.
Step identifiers and dependencies
Rule: walk_trail_simple_closed_separate_obligations · Uses: G001, S012