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Question 5774: Open-Sentence Binding and Truth Eligibility

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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R0010-05774 · q-20261008-advance-20261012-r03-proposition-0049

Original question and givens

In the classical integer context, consider x≤1 with assignment=12. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

  • G001 — In the classical integer context, consider x≤1 with assignment=12. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

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

bound integer comparison difference := sub(12, 1)

After

bound integer comparison difference = 11

Conditions: none

Substitute only after receiving an integer binding;the difference is x minus the threshold.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

x=12;threshold=1;x−threshold=11

After

bound statement truth=False

Conditions: none

A bound0 is a valid value too.The condition difference≤0 is exactly x≤threshold;programming-language truthiness is not mathematical binding.

Step identifiers and dependencies

Rule: bound_integer_formula_verdict · Uses: G001, S002

Reverse checks

  • 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 bound integer comparison difference 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

(Yes, False, 11)

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: c00521ee8bcc80c3aa8389d2d9de55371e9c00250a04c9ddf547af057228d838
  • candidate record SHA-256: a52116dc77ea40712f733709ac5d23804dd23a0be3c0e8930e87581b4b7c0bcf
  • semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
  • question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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