, Since S is a progressing and connected set of rules and, moreover (s, h) | = F (u), by Lemma 2, there exists a bijection ? : dom(h) ? nodes(u)\{?} such that, for all wi ? nodes(u), where i ? N, we have ? ?1 (wi) ? h(? ?1 (w)). Moreover, since (s, h) > t, there exists an injective mapping f : nodes(t) ? dom(h)\s(? ?{0, 1}) such that, for all w ? nodes(t) such that wi ? nodes(t), we have f (wi) ? h( f (w)). Consequently, we obtain the bijection ? ? f : nodes(t) ? nodes(u) \ {?} where, for all wi ? nodes(t), such that i ? N, there exists j ? N such that f (wi) = h( f (w)) j . Since ? is bijective and ? img(?)

, Since f is injective, h( f (w)) j = f (wi) f (w). Moreover, h( f (w)) j s(? ? {0, 1}) thus, by Fact 2, we obtain that f (w) = ? ?1 (v), as a consequence of h( f (w)) j = h(? ?1 (v)) m . Then we obtain that, for all wi ? nodes(t), By Lemma 2, there exists m ? N such that ? ?1 (vk) = h(? ?1 (v)) m

, By the definition of S, namely rule (32), there exists a unique node w 0 ? nodes(u) \ {?} such that u(w 0 ) = (c i , ? i ), for some formula ? i , where i ? {1, 2, 3}. Distinguishing the cases i = 1, 2, 3 and using the fact that (s, h) > t, one shows that t breaks the one of the conditions (I), (II) or (III), respectively, thus t is not a derivation of M. The proof is along the lines of the second point of Lemma 3

. Const, 0 ) as a sibling to a hat of height N, that occurs in c i (y 0 ), for all i = 1, 2, 3. Then the entailment p M (x) | = c M (x) holds if and only if, for each structure (s, h) such that (s, h) | = p M (x) and each extension s

, Lemma 5. The entailment p M (x) | = S c M (x) holds if and only if the membership problem (M, ) has a negative answer

, Since t is a derivation, it is also a pseudo-derivation of M and, by Lemma 3, there exists a structure (s, h) such that (s, h) | = S p M (x) and (s, h) t. Moreover, (s, h) > t follows from (s, h) t and

T. Antonopoulos, N. Gorogiannis, C. Haase, M. I. Kanovich, and J. Ouaknine, Foundations for decision problems in separation logic with general inductive predicates, Foundations of Software Science and Computation Structures -17th International Conference, FOSSACS 2014, Held as Part of the European Joint Conferences on Theory and Practice of Software, vol.8412, pp.411-425, 2014.

J. Brotherston, C. Fuhs, J. Pérez, and N. Gorogiannis, A decision procedure for satisfiability in separation logic with inductive predicates, Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS '14, vol.25, pp.1-25, 2014.

K. Ashok, D. Chandra, L. J. Kozen, and . Stockmeyer, Alternation. J. ACM, vol.28, issue.1, pp.114-133, 1981.

R. Iosif, A. Rogalewicz, and J. Simacek, The tree width of separation logic with recursive definitions, Proc. of CADE-24, vol.7898, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01418897

R. Iosif, A. Rogalewicz, and T. Vojnar, Deciding entailments in inductive separation logic with tree automata, Automated Technology for Verification and Analysis -12th International Symposium, vol.2014, pp.201-218, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01418889

S. Samin, . Ishtiaq, W. Peter, and . Hearn, Bi as an assertion language for mutable data structures, ACM SIGPLAN Notices, vol.36, pp.14-26, 2001.

J. Katelaan, C. Matheja, and F. Zuleger, Effective entailment checking for separation logic with inductive definitions, Tools and Algorithms for the Construction and Analysis of Systems -25th International Conference, TACAS 2019, Held as Part of the European Joint Conferences on Theory and Practice of Software, vol.11428, pp.319-336, 2019.