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T. the combination of effects [5]. Acknowledgement. We thank Erwin R. Catesbeiana for finite discussions about infinite iterations. : Computational types from a logical perspective. J. Funct. Prog. : A confluent reduction for the lambda-calculus with surjective pairing and terminal object. J. Funct. Program. : Linear regions are all you need. In: Sestoft, P. ) ESOP 2006. LNCS, vol. 3924, pp. 7–21. : Deriving backtracking monad transformers. : Combining effects: Sum and tensor. Theoret. Comput. Sci.

Xkm : T Akm ) ✄ a : T B occurring in p, q, whose return value is either bound to some variable xk or propagated to the top of the term. In the latter case, we just assume k = n + 1. Now put a ˆ(z) = do x ← a πk1 z, . . , πkn z ; ret π1 z, . . , πk−1 z, x, πk+1 , . . , πn+1 z . Having defined the interpretation of all the symbols a ˆ in this manner, we obtain an equation over the original signature Σ. It can be shown by induction that the original equation p = q can by obtained from it by application of the operator λt.

We next require some auxiliary machinery for additive monads, which as a side product induces a simple normalisation-based algorithm for deciding equality over additive monads. Consider the following rewriting system, inspired by [2]. (p : 1n ) fst(p), n n , snd(p) fst(p), snd(p) do x ← (p : T 1n ); ret n do x ← p; ret x n p p p p p fst p, q snd p, q do x ← ret p; q do x ← (do y ← p; q); r p q q[p/x] (∗) do x ←p; y ← q; r 26 S. Goncharov, L. Schr¨oder, and T. Mossakowski Basic monad laws: do x ← (do y ← p; q); r = do x ← p; y ← q; r (bind) (eta1 ) do x ← ret a; p = p[a/x] (eta2 ) do x ← p; ret x = p Extra axioms for nondeterminism: (plus∅) p+∅ =p (comm) p+q =q+p (idem) p+p=p (assoc) p + (q + r) = (p + q) + r (bind∅1 ) do x ← p; ∅ = ∅ (bind∅2 ) do x ← ∅; p = ∅ (distr1 ) do x ← p; (q + r) = do x ← p; q + do x ← p; r (distr2 ) do x ← (p + q); r = do x ← p; r + do x ← q; r Extra axioms and rules for Kleene star: (unf1 ) init x ← p in q ∗ = p + do x ← (init x ← p in q ∗ ); q (unf2 ) init x ← p in q ∗ = p + init x ← (do x ← p; q) in q ∗ (init) (ind1 ) init x ← (do y ← p; q) in r ∗ = do y ← p; init x ← q in r ∗ do x ← p; q ≤ p init x ← p in q ∗ ≤ p (ind2 ) (y ∈ / F V (r)) do x ← q; r ≤ r do x ← (init x ← p in q ∗ ); r ≤ do x ← p; r Fig.

### Algebraization of Hamiltonian systems on orbits of Lie groups

by Daniel

4.2