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### Abstract Modularity [chapter]

Michael Abbott, Neil Ghani, Christoph Lüth
2005 Lecture Notes in Computer Science
We now show that strong normalisation is monadic.  ...  Now observe that the free lifting M ∅ of the monad M is strongly normalising because the only rewrites in M ∅ are variable rewrites.  ...

### Monadic translation of classical sequent calculus

JOSÉ ESPÍRITO SANTO, RALPH MATTHES, KOJI NAKAZAWA, LUÍS PINTO
2013 Mathematical Structures in Computer Science
normalisation.  ...  Through strict simulation, the strong normalisation of simply typed λ-calculus is inherited by monadic λµ, and then by cbn and cbv λµμ, thus reproving strong normalisation in an elementary syntactical  ...  Since the optimised cbn CPS translation also preserves typability, we can infer the strong normalisation of λµμ n from the strong normalisation of λ[β v ]. A.3.2. Cbv case.  ...

### Monads and modular term rewriting [chapter]

Christoph Lüth, Neil Ghani
1997 Lecture Notes in Computer Science
This paper provides further support for monadic semantics of rewriting by giving a categorical proof of the most general theorem concerning the modularity of strong normalisation.  ...  Monads can be used to model term rewriting systems by generalising the well-known equivalence between universal algebra and monads on the category Set.  ...  Firstly, monads can be used to model more general notions of term rewriting for which current modularity results are less than satisfactory.  ...

### Monad Comprehensions: A Versatile Representation for Queries [chapter]

Torsten Grust
2004 The Functional Approach to Data Management
The monad comprehension normalisation rules provide an elegant proof of this transformation: select distinct f x from xs as x where p x z = {f x | x ← xs,p x z} = {f x | x ← xs,or [ |p x v | v ← z| ]}  ...  = {f x | x ← xs,v ← z,p x v} = {f x | x ← xs,v ←{ |g x y | y ← ys,q x y } |,p x v} = {f x | x ← xs,y ← ys,q x y,p x (g x y)} More Unnesting Monad comprehension normalisation readily unnests queries of  ...

### The marriage of effects and monads

1999 SIGPLAN notices
Here we marry effects to monads, uniting two previously separate lines of research.  ...  The same technique should allow one to transpose any effect system into a corresponding monad system.  ...  (τ, κ) whereω is a sequence of type, region, or effect variables; the scheme is normalised if τ is normalised.  ...

### The marriage of effects and monads

2003 ACM Transactions on Computational Logic
Here we marry effects to monads, uniting two previously separate lines of research.  ...  The same technique should allow one to transpose any effect system into a corresponding monad system.  ...  (τ, κ) whereω is a sequence of type, region, or effect variables; the scheme is normalised if τ is normalised.  ...

### The marriage of effects and monads

1998 Proceedings of the third ACM SIGPLAN international conference on Functional programming - ICFP '98
Here we marry effects to monads, uniting two previously separate lines of research.  ...  The same technique should allow one to transpose any effect system into a corresponding monad system.  ...  (τ, κ) whereω is a sequence of type, region, or effect variables; the scheme is normalised if τ is normalised.  ...

### Computational types from a logical perspective

P. N. BENTON, G. M. BIERMAN, V. C. V. DE PAIVA
1998 Journal of functional programming
We give natural deduction, sequent calculus and Hilbert-style presentations of this logic and prove strong normalisation and con uence results.  ...  Moggi's computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many di erent notions of computation have the categorical structure of a strong monad  ...  c into a precongruence and these all follow trivially from the compositional nature of the translation.Corollary 10 (Strong Normalisation) monad over a category C with nite products is a monad (T; ;  ...

### Algebras of Higher Operads as Enriched Categories

Michael Batanin, Mark Weber
2008 Applied Categorical Structures
We decribe the correspondence between normalised ω-operads in the sense of [1] and certain lax monoidal structures on the category of globular sets.  ...  Within the aforementioned correspondence, we provide also an equivalence between the algebras of a given normalised ωoperad, and categories enriched in globular sets for the induced lax monoidal structure  ...  A is normalised as a monad or endofunctor.  ...

### Algebras of higher operads as enriched categories [article]

Michael Batanin, Mark Weber
2008 arXiv   pre-print
We decribe the correspondence between normalised $\omega$-operads and certain lax monoidal structures on the category of globular sets.  ...  Within the aforementioned correspondence, we provide also an equivalence between the algebras of a given normalised $\omega$-operad, and categories enriched in globular sets for the induced lax monoidal  ...  A is normalised as a monad or endofunctor.  ...

### Interactive Programs in Dependent Type Theory [chapter]

Peter Hancock, Anton Setzer
2000 Lecture Notes in Computer Science
The inspiration is the 'I/O-monad' of Haskell.  ...  We present I/O-trees in two forms that we call 'non-normalising' and 'normalising'.  ...  A special case of a monad is the I/O-monad. When referring to the I/O-monad we write (IO A) instead of (M A). The interpretation of IO is as follows.  ...

### Reducibility and ⊤ ⊤-Lifting for Computation Types [chapter]

Sam Lindley, Ian Stark
2005 Lecture Notes in Computer Science
We propose -lifting as a technique for extending operational predicates to Moggi's monadic computation types, independent of the choice of monad.  ...  The method appears robust: we apply it to show strong normalisation for the computational metalanguage extended with sums, and with exceptions.  ...  simulate all definable monads [12] .  ...

### Monadic Translation of Intuitionistic Sequent Calculus [chapter]

José Espírito Santo, Ralph Matthes, Luís Pinto
2009 Lecture Notes in Computer Science
Indeed, strong normalisation follows immediately from strict simulation, since the target system is itself strongly normalising.  ...  Section 4 defines and proves the properties of the monadic translation and its optimized variant, and strong normalisation for λJ mse is obtained.  ...  Strong normalisation of λ[β, assoc, perm] will be needed below in Section 5.1 for translation F .  ...

### Backtracking with cut via a distributive law and left-zero monoids

MACIEJ PIRÓG, SAM STATON
2017 Journal of functional programming
We give two descriptions of the resulting monad: as the monad of free left-zero monoids, and as a composition via a distributive law of the list monad and the 'unary idempotent operation' monad.  ...  We employ the framework of algebraic effects to augment the list monad with the pruning cut operator known from Prolog.  ...  If R is confluent and normalising, then for all Σ T -terms t and s, it is the case that t ≈ T s if and only if nf(t) = nf(s).  ...

### A Lean Tactic for Normalising Ring Expressions with Exponents (Short Paper) [chapter]

Anne Baanen
2020 Lecture Notes in Computer Science
This paper describes the design of the normalising tactic ring exp for the Lean prover. This tactic improves on existing tactics by extending commutative rings with a binary exponent operator.  ...  The ring exp m monad contains a state monad transformer to keep track of which atoms are definitionally equal.  ...  The calculations of the eval function are thus done in an extension of the tactic monad, called the ring exp m monad.  ...
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