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Question 5736: Equality, Negation, and Signed Difference Witnesses

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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R0010-05736 · q-20261008-advance-20261012-r03-proposition-0061

Original question and givens

In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

  • G001 — In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

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

classical integer formulas with explicit binding/data

After

a truth verdict is not a delivered general proof object

Conditions: none

A false closed statement is still a proposition.These finite substitutions and comparison witnesses do not promote a truth label to a general formal proof or completeness of arbitrary inference rules.

Step identifiers and dependencies

Rule: proposition_truth_and_proof_scope · Uses: G001

S002 · calculation

Before

integer comparison difference0 := sub(-2, 0)

After

integer comparison difference0 = -2

Conditions: none

Compute a−b from the ordered pair,retaining zero and either sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

row=0;a=-2;b=0;a−b=-2

After

statement=a=b;truth=False

Conditions: none

Equality requires difference0 and ≤ requires a nonpositive difference;the signed difference supplies the exact comparison witness,not a candidate report.

Step identifiers and dependencies

Rule: closed_integer_statement_row · Uses: G001, S002

S004 · calculation

Before

integer comparison difference1 := sub(-9, 2)

After

integer comparison difference1 = -11

Conditions: none

Compute a−b from the ordered pair,retaining zero and either sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

row=1;a=-9;b=2;a−b=-11

After

statement=a=b;truth=False

Conditions: none

Equality requires difference0 and ≤ requires a nonpositive difference;the signed difference supplies the exact comparison witness,not a candidate report.

Step identifiers and dependencies

Rule: closed_integer_statement_row · Uses: G001, S004

S006 · calculation

Before

integer comparison difference2 := sub(0, 0)

After

integer comparison difference2 = 0

Conditions: none

Compute a−b from the ordered pair,retaining zero and either sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

row=2;a=0;b=0;a−b=0

After

statement=a=b;truth=True

Conditions: none

Equality requires difference0 and ≤ requires a nonpositive difference;the signed difference supplies the exact comparison witness,not a candidate report.

Step identifiers and dependencies

Rule: closed_integer_statement_row · Uses: G001, S006

S008 · implies

Before

[False, False, True]

After

[True, True, False]

Conditions: none

Take classical negation entrywise;a=b and a≠b are complementary.A false original statement does not cease to be a proposition.

Step identifiers and dependencies

Rule: classical_equality_negation · Uses: G001, S007

Reverse checks

  • 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 integer comparison difference2 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 integer comparison difference1 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 — Recompute integer comparison difference0 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

([False, False, True], [True, True, False], [-2, -11, 0])

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: ec49c006e3081c3648024936636f57aa2a31a7b5727338774733445422f4b3a5
  • candidate record SHA-256: 223f04c8d9226c068d829e06615f695e2a576e92153dbfaa6d92f1c417255637
  • semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
  • question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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