Bibliographic record
Abstract
Tilings are defined inductively. It is shown that one form of mutilated chess board cannot be tiled with dominoes, while another one can be tiled with L-shaped tiles. Sections 1 and 2 are by Paulson and described elsewhere [1]. Section 3 is by Nipkow and formalizes a well-known argument from the literature [2]. Please add further fun examples of this kind! theory Tilings imports Main begin 1 Inductive Tiling inductive-set tiling:: ′ a set set ⇒ ′ a set set for A:: ′ a set set where empty [simp, intro]: {} ∈ tiling A | Un [simp, intro]: [ a ∈ A; t ∈ tiling A; a ∩ t = {}] = ⇒ a ∪ t ∈ tiling A lemma tiling-UnI [intro]: [ t ∈ tiling A; u ∈ tiling A; t ∩ u = {} ] = ⇒ t ∪ u ∈ tiling A apply (induct set: tiling) apply (auto simp add: Un-assoc) done lemma tiling-Diff1E: assumes t−a ∈ tiling A and a ∈ A and a ⊆ t shows t ∈ tiling A proof − from assms(2 −3) have EX r. t = r Un a & r Int a = {} by (metis Diff-disjoint Int-commute Un-Diff-cancel Un-absorb1 Un-commute) thus?thesis using assms(1,2) by (auto simp:Un-Diff) (metis Compl-Diff-eq Diff-Compl Diff-empty Int-commute Un-Diff-cancel 1 qed Un-commute double-complement tiling.Un) lemma tiling-finite: assumes ∧ a. a ∈ A = ⇒ finite a shows t ∈ tiling A = ⇒ finite t apply (induct set: tiling) using assms apply auto done
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.003 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.005 | 0.012 |
| Open science | 0.002 | 0.006 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.080 | 0.026 |
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".