# Questions tagged [definition]

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### Defining a Recursive Function decreasing on one argument with < and another structurally

I want to define a function which decreases in one argument using < and on another structurally. What is the least painful way to do this? Among the many options (Function, Program Fixpoint, ...
2 votes
1 answer
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### How do clausal definitions work?

I'm most familiar with the metatheory of calculi based around expressions. But systems like Agda use separate clauses for definitions. ...
• 2,768
1 vote
1 answer
58 views

### Question about default definitions in fields

In Unclarity about Preorder class in Lean4 I asked why the third and fourth field (lt and lt_iff_le_not_le) in the definition of MyPreorder below would both be necessary, as one follows from the other ...
5 votes
2 answers
211 views

### Unclarity about Preorder class in Lean4

I realize the port of Mathlib to Lean4 is not finished yet, but I've run into a definition that I do not quite understand. I'm quite new at using theoremprovers as well as stackexchange, so please be ...
0 votes
1 answer
72 views

### Define a function using another function

I want to define a translation function g from another function f. Definition trans (f g : Q -> Q) (t : Q) := forall q : Q, f (q + t) == g q. In the above code,...
• 357
5 votes
0 answers
205 views

### An algorithm for the substitution of formulas for predicates in first order logic

I am trying to find a detailed description of the definition of substitution of formulas for predicates in first order logic and an implementation of this as a function in Lean or Haskell. The aim is ...
• 419
4 votes
2 answers
191 views

### Coq defining a hierarchy of collections of integers with infinitely many "levels"

I'm trying to formalize a small part of higher-order arithmetic in Coq as an exercise (Wikipedia article for second-order arithmetic). It's straightforward to formalize something resembling second-...
• 2,711