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• The unbounded fractional authoritative camera. · 151dda05
Robbert Krebbers authored
```The unbounded fractional authoritative camera is a version of the fractional
authoritative camera that can be used with fractions `> 1`.

Most of the reasoning principles for this version of the fractional
authoritative cameras are the same as for the original version. There are two
difference:

- We get the additional rule that can be used to allocate a "surplus", i.e.
if we have the authoritative element we can always increase its fraction
and allocate a new fragment.

✓ (a ⋅ b) → ●U{p} a ~~> ●U{p + q} (a ⋅ b) ⋅ ◯U{q} b

- At the cost of that, we no longer have the `◯U{1} a` is an exclusive
fragmental element (cf. `frac_auth_frag_validN_op_1_l`).```
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• A specific constructor for injecting values in expressions · 9646293e
Jacques-Henri Jourdan authored
```We add a specific constructor to the type of expressions for injecting
values in expressions.

- Values can be assumed to be always closed when performing
substitutions (even though they could contain free variables, but it
turns out it does not cause any problem in the proofs in
practice). This means that we no longer need the `Closed` typeclass
and everything that comes with it (all the reflection-based machinery
contained in tactics.v is no longer necessary). I have not measured
anything, but I guess this would have a significant performance
impact.

- There is only one constructor for values. As a result, the AsVal and
IntoVal typeclasses are no longer necessary: an expression which is
a value will always unify with `Val _`, and therefore lemmas can be
stated using this constructor.

Of course, this means that there are two ways of writing such a thing
as "The pair of integers 1 and 2": Either by using the value
constructor applied to the pair represented as a value, or by using
the expression pair constructor. So we add reduction rules that
transform reduced pair, injection and closure expressions into values.
At first, this seems weird, because of the redundancy. But in fact,
this has some meaning, since the machine migth actually be doing
something to e.g., allocate the pair or the closure.

These additional steps of computation show up in the proofs, and some
additional wp_* tactics need to be called.```
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