# Library test

Universe variables in X are global (X is not universe polymorphic). Therefore, 'Compare Universes "i" ? "j"' will output 'Inferred relation: i > j'.

Compare Universes "i" ? "j".

Polymorphic Definition PLX :

Type :=

Type.

As of Coq8.5~Beta3, universe of the form Top.n are not assumed to be all existing anymore.
Universe variables in PLX are local (PLX is universe polymorphic). Therefore, 'Compare Universes "Top.3" ? "Top.4"' does not make sense and results in an "Anomaly" error. This is because Top.3 and Top.4 do not exist in the global context.

In Coq8.5~beta2 and earlier all universes of the form Top.n were asumed to be existing at all times and so the previous command would have said that the given universe levels are unrelated.

Compare Universes "Top.3" ? "Top.4". – This causes an "Anomaly"

In this case, one can use 'Compare Universes "Top.3" ? "Top.4" Of PLX' which will output 'Inferred relation: Top.3 > Top.4'

Here we have to use 'test.3' and 'test.4' respecitvely for compilation reasons.
As of Coq8.5~Beta3, coqc and coqtop treat universe neames differently.
In coqtop anonymous universe levels are of the form Top.n while in coqc, they are of the form
M.n where M is the name of the module being compiled.

Hence the following command:

Compare Universes "test.3" ? "test.4" of PLX.

One can also ask if a particular relation holds within the global or local constraints.

Note that universe levels generated internally by Coq, e.g. "Top.3" above, are not fixed and can change. A better way to compare universe levels in a more consistent way is to use declared universe variables, e.g., "x" and "y" below.

Universe x y.

Polymorphic Definition W :=

PLX@{

x y}.

Here we have bound universe variables of PLX to "x" and "y" in the context of definition W. Therefore, 'Compare Universes "x" ? "y" Of W' now outputs 'Inferred relation: x > y'.

Compare Universes "x" ? "y" of W.

Compare Universes (acting both globally and locally) can also take into account a relation and check if that relation holds.

The following is a table of commands and their outputs:

Commad | Result |

Compare Universes "x" < "y" Of W. | Doesn't hold because: x > y |

Compare Universes "x" <= "y" Of W. | Doesn't hold because: x > y |

Compare Universes "x" = "y" Of W. | Doesn't hold because: x > y |

Compare Universes "x" > "y" Of W. | Holds: x > y |

Compare Universes "x" >= "y" Of W. | Holds because: x > y |

Fail Compare Universes "x" < "y" of W.

Fail Compare Universes "x" ≤ "y" of W.

Fail Compare Universes "x" = "y" of W.

Compare Universes "x" > "y" of W.

Compare Universes "x" ≥ "y" of W.

By default, Compare Unvierses command issues an error if the comparison result doesn't hold.
This feature can be dissibled by:

Unset Universe Comparison Error.

Compare Universes "x" < "y" of W.

And reenabled by:

Set Universe Comparison Error.

Fail Compare Universes "x" < "y" of W.