Abstract A proof of Bondy’s Theorem following Bollobas [1].
Bibliographic record
Abstract
begin lemma card-less-if-surj-not-inj: [ [ finite A; f ‘ A = B; ¬ inj-on f A]] = ⇒ card B < card A by (metis assms card-image-le inj-on-iff-eq-card order-le-neq-trans) theorem Bondy: assumes ∀A ∈ F. A ⊆ X and card X ≥ 1 and card F = card X shows ∃D. D ⊆ X & card D < card X & card (inter D ‘ F) = card F proof − from assms(2,3) have finite F and finite X by (metis card-infinite not-one-le-zero)+ { fix m have m < card F = ⇒ ∃D. D ⊆ X & card D ≤ m & card (inter D ‘ F) ≥ m + 1 proof (induction m) case 0 hence {} ⊆ X & card {} ≤ 0 & card (inter {} ‘ F) ≥ 0 + 1 by auto (metis Suc-leI card-eq-0-iff empty-is-image finite-imageI gr0I) thus ∃D. (D ⊆ X & card D ≤ 0 & card (inter D ‘ F) ≥ 0 + 1) by blast next case (Suc m) hence m < card F by arith with Suc.IH obtain D where D: D ⊆ X ∧ card D ≤ m ∧ m + 1 ≤ card (inter D ‘ F) by auto with 〈finite X 〉 have finite D by (auto intro: finite-subset) show?case proof (cases card (inter D ‘ F) = card F) case True hence D ⊆ X ∧ card D ≤ Suc m ∧ Suc m + 1 ≤ card(inter D ‘ F) using D Suc.prems by auto thus?thesis by blast next 1 case False hence ∼ inj-on (inter D) F by (auto simp: card-image) then obtain A1 A2 where A1 ∈ F and A2 ∈ F and
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.007 |
| Insufficient payload (model declined to judge) | 0.132 | 0.042 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".