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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.002 |
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 teacher head, 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".