feat(Order/Incomparable): incomparability relation

[vihdzp]

We define a relation for incomparable elements in a preorder and introduce basic API for it.

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• depends on: feat(Order/Antisymmetrization): AntisymmRel lemmas #19571

[github-actions]

PR summary 632eded3a0

Import changes exceeding 2%

%	File
+2.34%	Mathlib.Order.Antisymmetrization

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Order.Antisymmetrization	214	219	+5 (+2.34%)

Import changes for all files

Files	Import difference
Mathlib.Order.Antisymmetrization	5
Mathlib.Order.Incomparable (new file)	220

---

Declarations diff

+ AntisymmRel.antisymmRel_congr
+ AntisymmRel.antisymmRel_congr_left
+ AntisymmRel.antisymmRel_congr_right
+ AntisymmRel.comm
+ AntisymmRel.ge
+ AntisymmRel.incompRel_congr
+ AntisymmRel.incompRel_congr_left
+ AntisymmRel.incompRel_congr_right
+ AntisymmRel.le
+ AntisymmRel.le_congr
+ AntisymmRel.le_congr_left
+ AntisymmRel.le_congr_right
+ AntisymmRel.lt_congr
+ AntisymmRel.lt_congr_left
+ AntisymmRel.lt_congr_right
+ AntisymmRel.not_incompRel
+ AntisymmRel.trans_incompRel
+ AntisymmRel.trans_le
+ AntisymmRel.trans_lt
+ IncompRel
+ IncompRel.decidableRel
+ IncompRel.not_antisymmRel
+ IncompRel.not_ge
+ IncompRel.not_gt
+ IncompRel.not_le
+ IncompRel.not_lt
+ IncompRel.symm
+ IncompRel.trans_antisymmRel
+ LE.le.lt_or_antisymmRel
+ LE.le.not_incompRel
+ LE.le.trans_antisymmRel
+ LT.lt.not_incompRel
+ LT.lt.trans_antisymmRel
+ _
+ antisymmRel_compl
+ antisymmRel_iff
+ incompRel_compl
+ incompRel_iff
+ incompRel_of_antisymmRel_of_incompRel
+ incompRel_of_incompRel_of_antisymmRel
+ incompRel_refl
+ incompRel_swap
+ instance : @Trans α α α (AntisymmRel (· ≤ ·)) (IncompRel (· ≤ ·)) (IncompRel (· ≤ ·))
+ instance : @Trans α α α (AntisymmRel (· ≤ ·)) (· < ·) (· < ·)
+ instance : @Trans α α α (AntisymmRel (· ≤ ·)) (· ≤ ·) (· ≤ ·)
+ instance : @Trans α α α (IncompRel (· ≤ ·)) (AntisymmRel (· ≤ ·)) (IncompRel (· ≤ ·))
+ instance : @Trans α α α (· < ·) (AntisymmRel (· ≤ ·)) (· < ·)
+ instance : @Trans α α α (· ≤ ·) (AntisymmRel (· ≤ ·)) (· ≤ ·)
+ instance : IsSymm α (AntisymmRel r)
+ instance : IsSymm α (IncompRel r)
+ instance [IsIrrefl α r] : IsRefl α (IncompRel r)
+ instance [IsRefl α r] : IsRefl α (AntisymmRel r)
+ instance [IsTrans α r] : IsTrans α (AntisymmRel r)
+ le_iff_lt_or_antisymmRel
+ le_of_antisymmRel_of_le
+ le_of_le_of_antisymmRel
+ lt_of_antisymmRel_of_lt
+ lt_of_lt_of_antisymmRel
+ lt_or_antisymmRel_or_gt_or_incompRel
+ lt_or_eq_or_gt_or_incompRel
+ not_incompRel
+ not_le_iff_lt_or_incompRel

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• feat(Order/Antisymmetrization): AntisymmRel lemmas #19571
  By Dependent Issues (🤖). Happy coding!

[jsm28]

If you do want to use a different symbol, U+01C1 LATIN LETTER LATERAL CLICK ǁ and U+23F8 DOUBLE VERTICAL BAR ⏸ are also options to consider.