Elements of Quantitative Rewriting

We introduce a general theory of quantitative and metric rewriting systems, namely systems with relation enriched over quantales modelling abstract quantities. develop theories term-based refining cornerstone results (such as Newman’s Lemma, Church-Rosser Theorem, critical pair-like lemmas) to setting. To avoid distance trivialisation lack confluence issues, we non-expansive, linear term then generalise the latter novel class graded systems. These make modal context-sensitive, this way endowing coeffectful behaviours.