A proof procedure, in the spirit of the sequent calculus, is proposed to check the validity of entailments between Separation Logic formulas combining inductively defined predicates denoting structures of bounded tree width and theory reasoning. The calculus is sound and complete, in the sense that a sequent is valid iff it admits a (possibly infinite) proof tree. We also show that the procedure terminates in the two following cases: (i) When the inductive rules that define the predicates occurring on the left-hand side of the entailment terminate, in which case the proof tree is always finite. (ii) When the theory is empty, in which case every valid sequent admits a rational proof tree, where the total number of pairwise distinct sequents occurring in the proof tree is doubly exponential w.r.t. the size of the end-sequent.

本着序贯微积分的精神，提出了一种证明程序来检查分离逻辑公式之间蕴涵的有效性，该分离逻辑公式结合了表示有界树宽度结构的归纳定义谓词和理论推理。这个演算是健全和完整的，从某种意义上说，如果一个序列承认一个（可能是无限的）证明树，那么它就是有效的。我们还表明该过程在以下两种情况下终止：（i）当定义蕴涵左侧出现的谓词的归纳规则终止时，在这种情况下证明树总是有限的。(ii) 当理论为空时，在这种情况下，每个有效序列都承认一个有理证明树，其中证明树中出现的成对不同序列的总数相对于最终序列的大小是双指数的。