proof-and-counterexample

name	proof-and-counterexample
description	Prove, disprove, test, or repair mathematical statements by normalizing quantifiers and assumptions, building proof obligations, searching bounded domains safely, verifying counterexamples, and assigning evidence statuses that distinguish computation from proof.
license	MIT

Proof and Counterexample

Decision contract

Write the exact domain, quantifier order, hypotheses, conclusion, equality notion, edge cases, and intended evidence status before proof search. Write the exact negation. Decide whether the target is a general theorem, an exact finite proposition, or only a bounded test.

Check whether the statement is false before investing in a proof. Do not weaken a failed statement silently; create a repaired claim with its added hypothesis.

Reference usage

Read proof and counterexample evidence before counterexample-search, proof-obligation-authoring, or assigning proved-* or disproved. The reference contains exact domain declarations, expression limits, result statuses, and receipt rules.

Record material reference hashes. Preserve search and proof artifacts with $science-provenance, then use $science-review for any claim that leaves the local exercise context.

Workflow

1. Normalize the statement and negation with explicit types and quantifiers.
2. Attack omitted hypotheses, zero and empty cases, boundaries, singularities, small finite objects, and low dimensions.
3. When a bounded executable domain helps, run "<plugin-root>/scripts/python_runtime.sh" "<plugin-root>/scripts/search_counterexample.py". A returned assignment must satisfy every hypothesis and violate the conclusion.
4. If no counterexample is found, keep the status tested unless an exact finite domain was completely exhausted.
5. For a proof, choose direct, contrapositive, contradiction, induction, construction, invariant, extremal, or equivalence routes and decompose every material dependency into a proof-obligation graph.
6. Produce an informal-deductive, finite-exhaustion, or formal-kernel proof receipt appropriate to the evidence.
7. Cross-check statement hashes, assumptions, and unresolved obligations before changing the claim status.

Outputs

• mathematical-claim with statement hash, domain, assumptions, quantifiers, status, and permitted inference;
• counterexample-search receipt for bounded evaluation;
• proof-obligation-graph for deductive dependencies;
• proof-receipt identifying deductive, finite, formal, or computational evidence;
• repaired theorem and necessity/sufficiency discussion when the original statement fails;
• manifest and review receipt for material work.

Boundaries

• not proved is not false; no counterexample found is not proved.
• Random or floating-point samples cannot prove a universal statement.
• Exact finite exhaustion proves only the exact finite domain stated in the receipt.
• A symbolic computation is proof only when its transformations, domains, and theorem dependencies are justified.
• A passed obligation cannot depend on an open or failed obligation.
• A counterexample must satisfy every original hypothesis, not a weakened approximation.
• Stop and report conditional, tested, or unavailable rather than inventing a missing lemma or hiding a failed case.

Source basis

The proof discipline is independently synthesized from Lebl's Basic Analysis I and Hefferon's Linear Algebra; provenance and licenses are in ../../docs/TEXTBOOK_SOURCES.md.