Coq(co)Monads are used to encapsulate impure operations of a computation. A (co)monad is determined by an adjunction and further determines a specific type of adjunction called the (co)Kleisli adjunction. Mac Lane introduced the comparison theorem which allows comparing these adjunctions bridged ...
The coq standard library. http://coq.inria.fr/library/. [Jon75] K. A. De Jong. An analysis of the behavior of a class of genetic adaptive systems. PhD thesis, Ann Arbor, MI, USA, 1975. [Kot93] J. Kotowicz. Functions and finite sequences of real numbers. Journal of Formalized ...
The above numerical observations are not limited to this two-dimension example, and they are formalized for general LP in Sects. 3 and 4. Fig. 2 Plots to illustrate the geometry of the LP instance (5), and the numerical behaviors of PDHG for solving the LP instance with different ...
To get us to a Heyting algebra, we need more structure still — we need implication, which is like an internal function arrow, or an internal ≤ relation. Recall that the equation we want to satisfy is "c ∧ a ≤ b < -> c ≤ a → b". The idea is that we should be able to...
the structure we started with. However, at each individual leaf, the value would potentially be bottom. And, in fact, by standard reasoning (it takes an infinite amount of time to find the minimum of an infinite stream), we can conclude that whenrepMinis run on an infinite structure, ...
(bits): they allow for a more complex behavior which encompasses phenomena like coherent superposition, interference, or uncertainty relations. Yet, both classical and quantum bits can be formalized in a unified way that we now describe (for both single and multiple bits, i.e., circuits, we ...
\cref{axiom:univalence} is formalized in a similar fashion, too: % \begin{mathparpagebreakable} \inferrule*[right=$\UU_i$-\textsc{univ}] {\oftp\Gamma{A}{\UU_i} \\ \oftp\Gamma{B}{\UU_i}} {\oftp\Gamma{\univalence(A,B)}{\isequiv(\idtoeqv_{A,B})}} \end{mathparpagebre...
We present Pythagoras' proof of the irrationality of 2 both informal and formalized in (1) HOL, (2) Mizar, (3) PVS, (4) Coq, (5) Otter /Ivy, (6) Isabelle/Isar, (7) Alfa/Agda, (8) ACL2, (9) PhoX, (10) IMPS, (11) Metamath, (12) Theorema, (13) Lego, (14) Nuprl, ...
A, called an Abox, is a finite set of statements of the form C(a), R(a,b), a=b or a≠b. Note that we changed the standard definition of SHIQ by adding equalities of the form a=b for individuals a and b in the Abox. There are two reasons for adding equality between individu...
Towards Modeling and Verification of the CKB Block Synchronization Protocol in Coq. In Formal Methods and Software Engineering, Proceedings of the 22nd International Conference on Formal Engineering Methods, ICFEM 2020, Singapore, 1–3 March 2021; Springer: Cham, Switzerland, 2020; pp. 287–296. [...