Confused about the proof of Dijkstra Algorithm

In the proof of the correctness of Dijkstra algorithm, there is a lemma stating as follow:

Let u be v’s predecessor on a shortest path P:s->…->u->v from s to v. Then, If d(u) = δ(s,u) and edge (u, v) is relaxed, we have d(v) = δ(s,v), where the funciton δ(x, y) denotes the minimum path weight from x to y.

I wonder why we need the condition d(u) = δ(s,u) in this lemma. If Path P: s->…->u->v is a shortest path from s to v, then by the property of optimal substructure, the subpath s->…->u of P must also be a shortest path from s to u. Therefore, d(u) must equal to δ(s,u).

Does there exist the case that d(u) ≠ δ(s,u) but P: s->…->u->v is a shortest from s to v? If it does, can someone offer an example here.

Any help will be appreciated

PS: if you are interested in the entire proof. Check here, the proof starts at 45:30

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confused with this code [on hold]

Found this code while looking for comparisons and differences between imperative programming and functional programming languages in programming paradigms.

It was written that this below code is the recursion for functional programming.

(def (dup a) (con ((null ? a) (1) (else (cons (ara) dup(csd r a))))

 (def (dup a)                                                                 (con ((null? a) (1)            (else (cons (ara) dup(csd r a))) 

Is this code correct? If its correct what does it mean?

Have 3 domains and thinking to boost on google. But confused between the content

I bought 3 domains.

Example.com
Example.in
Example.com.au

if i add same content on all of three website. Then i want to rank these website in various countries like USA, India, and Australis with same keywords uses.

Is that content called plagiarized or i have to go with unique content on these three domains.

or simply i will go with <link rel="canonical"> What should i choose?

New and Confused

I am sorry in advance if I am wrong to post this thread here.
I am new here, in this community. I was looking for an appropriate place to post the introduction about myself. But, I don't know is it the right place or not.

Anyway, Hello everyone. I'm Monim. 25 years of age and a bit shy and nervous. So, that's all for today. Hoping for some welcoming replies.

Thank you.