Fixes new Coq master warning deprecated-hint-without-locality.

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.

Thanks to @simongregersen for reporting.

Note that prior to 0d7a7f06 we would get this instance automatically,
but that instance would not have the right computational behavior.

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

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

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

This allows for more control over `Hint Mode`.

This instance leads to exponential failing searches.

These trees are useful to show that other types are countable.

This patch was created using

  find -name *.v | xargs -L 1 awk -i inplace '{from = 0} /^From/{ from = 1; ever_from = 1} { if (from == 0 && seen == 0 && ever_from == 1) { print "Set Default Proof Using \"Type*\"."; seen = 1 } }1 '

and some minor manual editing

simplify_equality        => simplify_eq
simplify_equality'       => simplify_eq/=
simplify_map_equality    => simplify_map_eq
simplify_map_equality'   => simplify_map_eq/=
simplify_option_equality => simplify_option_eq
simplify_list_equality   => simplify_list_eq
f_equal'                 => f_equal/=

The /= suffixes (meaning: do simpl) are inspired by ssreflect.

Also, make our redefinition of done more robust under different
orders of Importing modules.

Also do some minor clean up.

Also, use a different encoding of lists.