2024年6月5日（水）

In proof assistants[edit]

Many proof assistants, such as Coq and Lean, define 1/0 = 0. This is due to the requirement that all functions are total. Such a definition does not create contradictions, as further manipulations (such as cancelling out) still require that the divisor is non-zero.^[33]^[34]

Tanter, Éric; Tabareau, Nicolas (2015), "Gradual certified programming in coq", DLS 2015: Proceedings of the 11th Symposium on Dynamic Languages, Association for Computing Machinery, arXiv:1506.04205, doi:10.1145/2816707.2816710, The standard division function on natural numbers in Coq, div, is total and pure, but incorrect: when the divisor is 0, the result is 0.
^ Buzzard, Kevin, "Division by zero in type theory: a FAQ", Xena Project (Blog), retrieved 2024-01-21

Division by zero - Wikipedia

en.wikipedia.org

Next, let's perform division by 0. In the case of select 1/0, an error message is returned as described above, but in the case of select div0null(1, 0), "0" is returned.

https://dev.classmethod.jp/articles/snowflake-function-div0null/

DIV0

Performs division like the division operator (/), but returns 0 if the divisor is 0 (instead of reporting an error).

https://docs.snowflake.com/ja/sql-reference/functions/div0

DIV0NULL

Performs division like the division operator (/), but returns 0 if the divisor is 0 or NULL

(rather than reporting an error or returning NULL).

https://docs.snowflake.com/ja/sql-reference/functions/div0null

https://www.ibm.com/docs/ja/i/7.3?topic=errors-handling-divide-by-zero

2024.4.20

There are signs of zero division adoption 1/0=0 in Microsoft Excel.

Below is the part that displays 0 or "no value" instead of #DIV/0!. It is common knowledge in the mathematical world that one should not think about division by zero, but when division by zero appeared, inconvenient situations occurred, such as errors, no solutions, and calculators stopping. In recent years, 1/0=0 has become widely adopted. Dividing by zero returns zero, which is true in strict mathematics for extended fractions in its natural sense, but it has become popular because it is convenient. In terms of meaning, it has become widely known that dividing by zero should not be considered or is impossible, and that in such cases it is better to express it as zero. Discovering the meaning of zero. Coq, Lean, IBM, etc. have a deeper understanding and use division by zero. Microsoft Excel seems to be weak because it is convenient. 2024.4.20.11:35

Evaluate 0 or no value in denominator

The easiest way to avoid seeing the #DIV/0! error is to use the IF function to evaluate for the presence of the denominator.

If 0 or no value, display 0 or "no value" instead of #DIV/0! as the result of the formula,

Otherwise, calculate the formula.

For example, if the formula that returns an error is =A2/A3, use =IF(A3,A2/A3,0) to return 0, and =IF(A3,A2/A3, ””) is used. You can also display your own message using =IF(A3,A2/A3,"Input required"). Using the QUOTIENT function from the first example, it would look like =IF(A3,QUOTIENT(A2,A3),0). This formula tells Excel to: IF (return the result of the formula if A3 exists, otherwise ignore the result).

https://support.microsoft.com/ja-jp/office/%E3%82%A8%E3%83%A9%E3%83%BC%E5%80%A4-div-0-%E3%82%92%E4%BF%AE%E6%AD%A3%E3%81%99%E3%82%8B%E6%96%B9%E6%B3%95-3a5a18a9-8d80-4ebb-a908-39e759a009a5

Lean is a programming language and proof support system (English version).

Based on the Calculus of constructions with induction.

Commentary[edit]

The Lean project is an open source project hosted on GitHub.

It was launched in 2013 by Leonardo de Moura of Microsoft Research [1].https://ja.wikipedia.org/wiki/Lean_(%E8%A8%BC%E6%98%8E%E3%82%A2%E3%82%B7%E3%82%B9%E3%82%BF%E3%83%B3%E3%83%88)

The Coq Proof Assistant: Welcome!

Coq is a proof support system. The core of Coq uses the programming language Gallina. The PI.R2 team of the French National Institute for Informatics and Automation (located within the PPS Institute) collaborated with the Ecole Polytechnique, the French National Institute of Crafts, the University of Paris VII, and the University of Paris XI (formerly the Ecole Normale Supérieure of Lyon). We are developing it jointly. Hugo Herbelin is the de facto head of development.

https://ja.wikipedia.org/wiki/Coq

The Isabelle^[a] 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. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle Archive of Formal Proofs (Isabelle AFP)^[2]

https://en.wikipedia.org/wiki/Isabelle_(proof_assistant)

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Institute of Reproducing Kernels

色々な事を書きます。マイペースで書きます。

2024年6月5日（水）

In proof assistants[edit]

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