CoqCoq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a certified program from the constructive proof of its formal specification. Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions. Coq is not an automated theorem prover but includes automatic theorem proving tactics (procedures) and various decision procedures.
Agda (programming language)Agda is a dependently typed functional programming language originally developed by Ulf Norell at Chalmers University of Technology with implementation described in his PhD thesis. The original Agda system was developed at Chalmers by Catarina Coquand in 1999. The current version, originally known as Agda 2, is a full rewrite, which should be considered a new language that shares a name and tradition. Agda is also a proof assistant based on the propositions-as-types paradigm, but unlike Coq, has no separate tactics language, and proofs are written in a functional programming style.
OCamlOCaml (oʊˈkæməl , formerly Objective Caml) is a general-purpose, high-level multi-paradigm programming language which extends the Caml dialect of ML with object-oriented features. OCaml was created in 1996 by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Didier Rémy, Ascánder Suárez, and others. The OCaml toolchain includes an interactive top-level interpreter, a bytecode compiler, an optimizing native code compiler, a reversible debugger, and a package manager (OPAM).
F* (programming language)F* (pronounced F star) is a functional programming language inspired by ML and aimed at program verification. Its type system includes dependent types, monadic effects, and refinement types. This allows expressing precise specifications for programs, including functional correctness and security properties. The F* type-checker aims to prove that programs meet their specifications using a combination of SMT solving and manual proofs. Programs written in F* can be translated to OCaml, F#, and C for execution.
Proof assistantIn computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.
Isabelle (proof assistant)The Isabelle automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring yet supporting explicit proof objects. Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods.
Functional programmingIn computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that map values to other values, rather than a sequence of imperative statements which update the running state of the program. In functional programming, functions are treated as first-class citizens, meaning that they can be bound to names (including local identifiers), passed as arguments, and returned from other functions, just as any other data type can.
EcosystemAn ecosystem (or ecological system) consists of all the organisms and the physical environment with which they interact. These biotic and abiotic components are linked together through nutrient cycles and energy flows. Energy enters the system through photosynthesis and is incorporated into plant tissue. By feeding on plants and on one another, animals play an important role in the movement of matter and energy through the system. They also influence the quantity of plant and microbial biomass present.
Lean (proof assistant)Lean is a theorem prover and programming language. It is based on the calculus of constructions with inductive types. The Lean project is an open-source project hosted on GitHub. It was launched by Leonardo de Moura at Microsoft Research in 2013. Lean has an interface, implemented as a Visual Studio Code extension and Language Server Protocol server, that differentiates it from other interactive theorem provers. It has native support for Unicode symbols, which can be typed using LaTeX-like sequences, such as "\times" for "×".
Military tacticsMilitary tactics encompasses the art of organizing and employing fighting forces on or near the battlefield. They involve the application of four battlefield functions which are closely related – kinetic or firepower, mobility, protection or security, and shock action. Tactics are a separate function from command and control and logistics. In contemporary military science, tactics are the lowest of three levels of warfighting, the higher levels being the strategic and operational levels.
Ecosystem serviceEcosystem services are the many and varied benefits to humans provided by the natural environment and healthy ecosystems. Such ecosystems include, for example, agroecosystems, forest ecosystem, grassland ecosystems, and aquatic ecosystems. These ecosystems, functioning in healthy relationships, offer such things as natural pollination of crops, clean air, extreme weather mitigation, and human mental and physical well-being.
SimplicitySimplicity is the state or quality of being simple. Something easy to understand or explain seems simple, in contrast to something complicated. Alternatively, as Herbert A. Simon suggests, something is simple or complex depending on the way we choose to describe it. In some uses, the label "simplicity" can imply beauty, purity, or clarity. In other cases, the term may suggest a lack of nuance or complexity relative to what is required. The concept of simplicity is related to the field of epistemology and philosophy of science (e.
Performance engineeringPerformance engineering encompasses the techniques applied during a systems development life cycle to ensure the non-functional requirements for performance (such as throughput, latency, or memory usage) will be met. It may be alternatively referred to as systems performance engineering within systems engineering, and software performance engineering or application performance engineering within software engineering.
PerformanceA performance is an act of staging or presenting a play, concert, or other form of entertainment. It is also defined as the action or process of carrying out or accomplishing an action, task, or function. In the work place, job performance is the hypothesized conception or requirements of a role. There are two types of job performances: contextual and task. Task performance is dependent on cognitive ability, while contextual performance is dependent on personality.
Marine coastal ecosystemA marine coastal ecosystem is a marine ecosystem which occurs where the land meets the ocean. Marine coastal ecosystems include many very different types of marine habitats, each with their own characteristics and species composition. They are characterized by high levels of biodiversity and productivity. For example, estuaries are areas where freshwater rivers meet the saltwater of the ocean, creating an environment that is home to a wide variety of species, including fish, shellfish, and birds.
Genetic programmingIn artificial intelligence, genetic programming (GP) is a technique of evolving programs, starting from a population of unfit (usually random) programs, fit for a particular task by applying operations analogous to natural genetic processes to the population of programs. The operations are: selection of the fittest programs for reproduction (crossover), replication and/or mutation according to a predefined fitness measure, usually proficiency at the desired task.
Computer performanceIn computing, computer performance is the amount of useful work accomplished by a computer system. Outside of specific contexts, computer performance is estimated in terms of accuracy, efficiency and speed of executing computer program instructions. When it comes to high computer performance, one or more of the following factors might be involved: Short response time for a given piece of work. High throughput (rate of processing work). Low utilization of computing resource(s). Fast (or highly compact) data compression and decompression.
Lake ecosystemA lake ecosystem or lacustrine ecosystem includes biotic (living) plants, animals and micro-organisms, as well as abiotic (non-living) physical and chemical interactions. Lake ecosystems are a prime example of lentic ecosystems (lentic refers to stationary or relatively still freshwater, from the Latin lentus, which means "sluggish"), which include ponds, lakes and wetlands, and much of this article applies to lentic ecosystems in general. Lentic ecosystems can be compared with lotic ecosystems, which involve flowing terrestrial waters such as rivers and streams.
Line segmentIn geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as ).
Montane ecosystemsMontane ecosystems are found on the slopes of mountains. The alpine climate in these regions strongly affects the ecosystem because temperatures fall as elevation increases, causing the ecosystem to stratify. This stratification is a crucial factor in shaping plant community, biodiversity, metabolic processes and ecosystem dynamics for montane ecosystems. Dense montane forests are common at moderate elevations, due to moderate temperatures and high rainfall.
