Why is the proof complete after proving only for one induction when we have more than one variable?

So I’m self-learning coq. And I came across the proof for associativity in addition forall (a b c : nat)

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Appearntly when we do induction a. after intros a b c., it creates 2 subgoals

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and afterwards we simply need to show that the two sides in both subgoals are equivalent, and the proof is completed.

So I’m just wondering why we don’t need to do induction b. and induction c. to complete the proof? Why only performing induction on a is able to complete the proof?

Or in other words, how come in the function that returns the proof, we just get b and c for “free”? Constructively don’t we need something like a double induction applied twice?

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If a Tempest cleric uses the Wrath of the Storm and Thunderbolt Strike features to push an attacker away, can the attacker complete its multiattack?

The Tempest Domain cleric’s Thunderbolt Strike feature (PHB, p. 62) says:

At 6th level, when you deal lightning damage to a Large or smaller creature, you can also push it up to 10 feet away from you.

I envision the Thunderbolt Strike as throwing the target creature away from it, not gently pushing it away. I mean, lightning is an instantaneous thing, and 10 feet is more than just losing your balance.

The Thunderbolt Strike feature of a Tempest cleric leaves me with many questions. For instance, if a monster is able to make a multiattack consisting of 2 claw attacks and then a bite, and it hits me with its first attack, I can use Wrath of Storm as a reaction to deal lightning damage to it, and thereby blast the creature back 10 feet using Thunderbolt Strike.

If the monster does not have 10 feet of movement left after being pushed, does it lose its other 2 attacks against me (if no other targets are in range of it)? Or does it get to make all 3 attacks before it is blasted away from me?

How is the Longest Path Problem NP complete?

From the following link:

https://www.csie.ntu.edu.tw/~lyuu/complexity/2016/20161129s.pdf

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So basically, in our iff proof, we have to show two directions:

Forward: If Hamiltonian Path has a yes-instance, so does longest path. This makes sense because we can just let “k” = |V| – 1 if hamiltonian path is yes. Then clearly there is a longest simple path with |V| – 1 edges.

I’m having trouble with the backward part

Backward: If Longest Path has a yes instance, so does longest path. Let’s assume that there is a longest path from s to t of length k (this can be a different k than the one we defined above?). How does that guarantee that there is a hamiltonian path from s-t? If it is the same k, where k = |V| – 1, then I agree there is a hamiltonian path, but what if this k is something different?

What is the complete lifecycle of souls in the Nine Hells?

The Descent into Avernus adventure provides more information about what happens to souls in the Nine Hells/Baator than was previously available (soul coins and their uses, in particular).

It’s also made it clear that I really don’t understand the paths that souls in the Hells can follow, and thus don’t feel able to convey the moral/logistical implications of any particular action relating to souls to players.

I’m looking for a complete picture of the soul lifecycle in the Hells (and I’m not expecting it to be linear). I could ask a bunch of smaller questions, but that wouldn’t actually help me understand the lifecycle as a whole so I can reason about it in unusual situations.

Answers should probably include:

  • States such as: being a living lawful evil mortal who hasn’t made a devilish pact, being a living mortal who has sold their soul, being a lemure, being in the River Styx, being tortured to draw out soul energy, being a higher rank devil, being trapped in a soul coin
  • Transitions such as: dying, being forged into a soul coin, being utterly annihilated, being freed from a soul coin, being promoted, being demoted
  • How the states and transitions relate to each other

5e materials & designer statements should take precedence, but materials from earlier editions are welcome in answers.

How to use Scrapebox as a complete beginner without using an API?

I have purchased as much as I can in the starting up of backlinking and I have followed advice from the best of them. I have bought Scrapebox, GSA, and The Best Spinner and Now I am out of money until next week.

Can anybody here guide me to using Scrapebox the best or only way possible without needing an api (just until next week) in conjunction with GSA Search Engine Ranker?

I have gsa set up and im just trying to get some lists because apparently my campaign has stopped because of that (it needs more urls to post to).

Can anyone tell me, am I able to extract Urls from Scrapebox and move forward with GSA just for the time being and if so can you tell me how?

Regards,

Michael