I have a beginner's question about Isabelle/HOL:
I want to prove the following lemma:
lemma
shows "{(x,y) . x ∈ {0..<n} ∧ y ∈ {0..<n} ∧ x = y} = {(x,x). x < n}"
But the proof state is:
proof (prove)
goal (1 subgoal):
1. {(x, y). x ∈ {0..<n} ∧ y ∈ {0..<n} ∧ x = y} = {(xa, x). x < n}
Why did the xa appear and how can I define the set the right (succinct) way?
The (x,y) in the set comprehension {(x,y). ....} is binding variable names. As you write {(x,x). x < n}, you bind two variables named x, where the second x shadows the first.
{(x,x). x < n} is just a nice syntax for a lambda term, actually. Internally, it translates to Collect (case_prod (λx. λx. x < n)). Looking at the term this way, the shadowing is more obvious.
To fix your problem, you have to explicitly express the information that the first and the second bound variable are to be identical, that is: {(x1,x2). x1 = x2 ∧ x1 < n}.
As a side note: The lemma you are trying to show is not true. (For example, n could be an int.) If you want n to be a nat, you have to make this explicit, for example by giving a type in your goal like this {(x,y). x ∈ {0..<(n::nat)} ∧ y ∈ {0..<n} ∧ x = y} = {(x1,x2). x1 = x2 ∧ x1 < n}.
Especially, if you are a beginner, I would strongly suggest to introduce free variables in lemma heads explicitly with the syntax lemma Name: fixes n :: ‹nat› assumes ‹...› shows ‹...›.
