Moreover, and suitable lemmas.

This follows the standard naming for Coq numbers, and avoids one using
the old lemma to break abstraction accidentally.

- Remove the coercion from `Qp` to `Qc`, and remove it into from `Qp_car` into
  `Qp_to_Qc` to be consistent with other conversion functions.
- Use `let '(..) = ...` in the definitions of `Qp_plus`/`Qp_mult`/`Qp_div` to
  avoid Coq tactics (like `injection`) to unfold these definitions eagerly.
- Define orders `Qp_le` and `Qp_lt`, instead of relying on the orders on `Qc`,
  which were obtained through the removed coercion into `Qc`.
- Lift lemmas about the orders from `Qc` to `Qp`.
- Improve variable names and use of notation scopes.

Tej Chajed authored 4 years ago and Robbert Krebbers committed 4 years ago

Set Default Goal Selector "!" makes it illegal to ever apply a tactic
with more than one goal (instead, must focus with bullets or braces).

Code not affected by a00d9bd8.

All occurrences are gone, except for one in `base.v` where you'd need different
functions.

However, I'm unsure this is an improvement: in lots of cases here, the function
didn't need to be guessed, but could be deduced by "simple" higher-order
unification, the one where unifying `?f ?a` against `g args last_arg` sets `?f =
g args`.

This was done using `sed --in-place 's/[[:space:]]\+$//' theories/*.v`.

This follows iris/iris!387
This closes issue #54.

For example, change `(!! i)` into `(.!! x)` so that `!!` can also be used as
a prefix, as done in VST for example.

This closes issue #42.

I have used the `sed` script below. This script took care of nearly all uses
apart from a few occurrences where a space was missing, e.g. `(,foo)`. In
this case, `coqc` will just fail, allowing one to patch up things manually.

The script is slightly too eager on Iris developments, where it also replaces
`($ ...)` introduction patterns. When porting Iris developments you thus may
want to remove the line for `$`.

```
sed '
s/(= /(.= /g;
s/ =)/ =.)/g;
s/(≠ /(.≠ /g;
s/ ≠)/ ≠.)/g;
s/(≡ /(.≡ /g;
s/ ≡)/ ≡.)/g;
s/(≢ /(.≢ /g;
s/ ≢)/ ≢.)/g;
s/(∧ /(.∧ /g;
s/ ∧)/ ∧.)/g;
s/(∨ /(.∨ /g;
s/ ∨)/ ∨.)/g;
s/( /(. /g;
s/ )/ .)/g;
s/(→ /(.→ /g;
s/ →)/ →.)/g;
s/($ /(.$ /g;
s/(∘ /(.∘ /g;
s/ ∘)/ ∘.)/g;
s/(, /(., /g;
s/ ,)/ ,.)/g;
s/(∘ /(.∘ /g;
s/ ∘)/ ∘.)/g;
s/(∪ /(.∪ /g;
s/ ∪)/ ∪.)/g;
s/(⊎ /(.⊎ /g;
s/ ⊎)/ ⊎.)/g;
s/(∩ /(.∩ /g;
s/ ∩)/ ∩.)/g;
s/(∖ /(.∖ /g;
s/ ∖)/ ∖.)/g;
s/(⊆ /(.⊆ /g;
s/ ⊆)/ ⊆.)/g;
s/(⊈ /(.⊈ /g;
s/ ⊈)/ ⊈.)/g;
s/(⊂ /(.⊂ /g;
s/ ⊂)/ ⊂.)/g;
s/(⊄ /(.⊄ /g;
s/ ⊄)/ ⊄.)/g;
s/(∈ /(.∈ /g;
s/ ∈)/ ∈.)/g;
s/(∉ /(.∉ /g;
s/ ∉)/ ∉.)/g;
s/(≫= /(.≫= /g;
s/ ≫=)/ ≫=.)/g;
s/(!! /(.!! /g;
s/ !!)/ !!.)/g;
s/(⊑ /(.⊑ /g;
s/ ⊑)/ ⊑.)/g;
s/(⊓ /(.⊓ /g;
s/ ⊓)/ ⊓.)/g;
s/(⊔ /(.⊔ /g;
s/ ⊔)/ ⊔.)/g;
s/(:: /(.:: /g;
s/ ::)/ ::.)/g;
s/(++ /(.++ /g;
s/ ++)/ ++.)/g;
s/(≡ₚ /(.≡ₚ /g;
s/ ≡ₚ)/ ≡ₚ.)/g;
s/(≢ₚ /(.≢ₚ /g;
s/ ≢ₚ)/ ≢ₚ.)/g;
s/(::: /(.::: /g;
s/ :::)/ :::.)/g;
s/(+++ /(.+++ /g;
s/ +++)/ +++.)/g;
' -i $(find -name "*.v")
```

Use Open/Close Scope without Local (i.e., export the scope opening)
only when the scope corresponds to the main purpose of the module.

Otherwise type class search ocasionally unfolds them and finds
wrong instances. Based on an issue reported by @jihgfee.

Jakob Botsch Nielsen authored 6 years ago and Robbert Krebbers committed 6 years ago

This changes the encoding used for finite lists of values of countable
types to be linear instead of exponential. The encoding works by
duplicating bits of each element so that 0 -> 00 and 1 -> 11, and then
separating each element with 10. The top 1-bits are not kept since we
know a 10 is starting a new element which ends with a 1.

Fix #28

Adding a hint without a database now triggers a deprecation warning in
Coq master (https://github.com/coq/coq/pull/8987).