by definition

Examples of definition

definition
Here we just recall the basic definitions to prevent confusion.
Such definitions are problematic in at least three ways.
To fix ideas and put the mathematical background in place, we start with some formal definitions.
The proof is a routine diagram paste based on the definitions.
We will begin by introducing a few basic definitions.
The results show that both definitions give the same parametric dependence.
Moreover, with such definitions, equational reasoning is not easy and leads to very large proof terms.
Then we check that our definitions are well-given.
The only non-trivial case concerns the operation of functional composition; proofs for the other high-level operations can be obtained easily by applying the semantic definitions.
Before proceeding, we will recall some definitions from lattice theory.
On the rare occasions where we present definitions out of sequence, we will explain what we are doing.
The purpose of these expressions is to give us some additional control over the evaluation of subexpressions when writing definitions of the above form.
The induction base is the first case in both definitions.
To introduce this kind of domain, we need the following definitions.
A few salient definitions and facts for general categories will be provided now.
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