Question 13192: The Norm Triangle Inequality Supplies the Metric Triangle Bound

Original question and givens

Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[4, -1], y=[3, -4], z=[-4, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).

• G001 — Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[4, -1], y=[3, -4], z=[-4, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).

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-norm-induced-metric

Step-by-step derivation

x=[4, -1];y=[3, -4];z=[-4, -1];same norm=l1

x-z=(x-y)+(y-z);universal norm triangle gives the metric triangle bound

x to z difference coordinate0 := sub(4, -4)

x to z difference coordinate0 = 8

x to z difference coordinate1 := sub(-1, -1)

x to z difference coordinate1 = 0

x to z:norm=l1;left=[4, -1];right=[-4, -1]

signed left-right=[8, 0];use same named norm

x to z:coordinate=0;signed value=8

absolute value=8

x to z:coordinate=1;signed value=0

absolute value=0

x to z l1 sum := sum(8, 0)

x to z l1 sum = 8

x to y difference coordinate0 := sub(4, 3)

x to y difference coordinate0 = 1

x to y difference coordinate1 := sub(-1, -4)

x to y difference coordinate1 = 3

x to y:norm=l1;left=[4, -1];right=[3, -4]

signed left-right=[1, 3];use same named norm

x to y:coordinate=0;signed value=1

absolute value=1

x to y:coordinate=1;signed value=3

absolute value=3

x to y l1 sum := sum(1, 3)

x to y l1 sum = 4

y to z difference coordinate0 := sub(3, -4)

y to z difference coordinate0 = 7

y to z difference coordinate1 := sub(-4, -1)

y to z difference coordinate1 = -3

y to z:norm=l1;left=[3, -4];right=[-4, -1]

signed left-right=[7, -3];use same named norm

y to z:coordinate=0;signed value=7

absolute value=7

y to z:coordinate=1;signed value=-3

absolute value=3

y to z l1 sum := sum(7, 3)

y to z l1 sum = 10

triangle right hand sum := sum(4, 10)

triangle right hand sum = 14

Reverse checks

• S020 · premise_and_result_audit — Recompute triangle right hand sum with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute y to z l1 sum with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute y to z difference coordinate1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute y to z difference coordinate0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute x to y l1 sum with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute x to y difference coordinate1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute x to y difference coordinate0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute x to z l1 sum with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute x to z difference coordinate1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute x to z difference coordinate0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(l1, 8, 4, 10, 14, Yes, norm_triangle_inequality)

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.