The first thing I would say about modal verbs is that their usage extends over many different contexts and so any general rules are bound to fail when taken out of context.
Outline
- Use of 'will' and 'would' to express belief
- Other uses
- The answer
1. Use of 'will' and 'would' to express belief
(Credit for this section should go to @snailboat)
To illustrate the difference in use of 'will' and 'would' for expressing certainty, snailboat suggested the following sentences at the ELL chat room:
1A. I'm sure they have been looking for those bank robbers.
1B. I'm sure they will have been looking for those bank robbers.
2A. I'm sure they had been looking for those bank robbers.
2B. I'm sure they would have been looking for those bank robbers.
In sentences 1A (they have been) and 2A (they had been) the speaker knows they have/had been looking for those bank robbers. The main difference is that the use of present perfect in 1A (they have been) indicates that the search has lasted at least until now.
In sentences 1B (they will have been) and 2B (they would have been) the speaker expresses a belief:
in sentence 1B (they will have been), the speaker expresses the belief that they have been looking for those bank robbers.
in sentence 2B (they would have been) the speaker expresses the belief that they had been looking for those bank robbers.
2. Other uses
Again, a word of caution, the interpretation above is not unique and can change if the context changes. This section shows other possible uses of will have been and would have been.
3. The answer
Let's now consider the example in your question:
A： We saw a police helicopter yesterday morning.
B： I'm sure they would have been looking for those bank robbers.
To be able to use "will have been" is necessary to change the context as described in the first section of this answer:
A： We have seen a police helicopter.
B： I'm sure they will have been looking for those bank robbers.
First, it is spelled as "Proceed" and the past tense is "Proceeded" which means "to go on" or "to continue".
As Collins says:
proceed (v.) to advance or carry on, esp after stopping
So, in the sentence you quote:
We procedded to Mughal Sarai to meet a retired chemical professor
it sounds like a group of people are on some kind of journey or adventure, and the next step of the trek was to travel to Mughal Sarai.
Best Answer
Note that "run the gamut" begins with the word "run". "Run" is a verb and normally requires a subject. While "run the gamut" is a specific idiom, it does not break this rule.
Normally in English the subject comes before the verb. So we would expect to see "[subject] runs the gamut ...", i.e. some subject coming before the word "run". Like, "Posters on ELL run the gamut from Americans to Britons to Indians to people from countries where English is not commonly spoken at all."
That said, English word order is not totally inflexible. You can write sentences that vary it, usually for emphasis or poetic style. For example, "Running the gamut from Americans to Japanese, posters on ELL are a diverse group."
Imperative sentences have an implied subject of "you", and so often begin with a verb. Like, "Run away!" But an imperative sentence using "run the gamut" seems unlikely.
Your example sentence doesn't make much sense to me. Who or what is running the gamut here? It is not the subject of the second clause, "laws". Laws do not run the gamut from nomadic tribes to skyscraper cities. Those aren't kinds of laws. The sentence mentions "societies". Those could be kinds of societies.
I think the writer meant something like, "In societies running the gamut from nomadic tribes to skyscraper cities, laws were central pillars in making them function."