feat(Data/Fintype/Card): existsUnique_notMem_image_of_injective_of_card_succ

[cjrl]

This pull requests adds a small theorem existsUnique_notMem_image_of_injective_of_card_succ to Mathlib/Data/Fintype/Card that says given an injective map f : α → β such that β has cardinality one more than α, there exists a unique element of β not in the image of f.

This can be viewed as going in the opposite direction of card_lt_of_injective_of_notMem.

This little fact is needed for our Latin Square PR #36698.

---

[github-actions]

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[github-actions]

PR summary 03d14ad0b5

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ card_eq_one_iff_existsUnique
+ existsUnique_mem_codomain_notMem_image_of_injective_of_card_eq_add_one
+ existsUnique_notMem_image_of_injective_of_card_eq_add_one

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 03d14ad).

• +3 new declarations
• −0 removed declarations

+Finset.card_eq_one_iff_existsUnique
+Finset.existsUnique_mem_codomain_notMem_image_of_injective_of_card_eq_add_one
+Fintype.existsUnique_notMem_image_of_injective_of_card_eq_add_one

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 03d14ad0b5
Reference commit 859caf703c

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/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).

[cjrl]

+t-set-theory

[ghseeli]

-awaiting-author

[dagurtomas]

Here is a slightly better approach, that results in more usable lemmas:

theorem card_image_compl_of_injective [DecidableEq β]
    (f : α → β) (hf : f.Injective) : #(univ \ univ.image f) = card β - card α := by
  simp [card_sdiff, card_image_of_injective _ hf]

/-- Given an injective map `f : α → β` such that `β` has cardinality one more
than `α`, there exists a unique element of `β` not in the image of `f`. -/
theorem existsUnique_notMem_image_of_injective_of_card_eq_add_one [DecidableEq β]
    (f : α → β) (hf : f.Injective) (h : card β = card α + 1) : ∃! x, x ∉ univ.image f := by
  have : #(univ \ univ.image f) = 1 := by grind [card_image_compl_of_injective]
  suffices ∃! x, x ∈ univ \ univ.image f by convert this; grind
  sorry -- prove a lemma saying `#s = 1 ↔ ∃! x, x ∈ s` for finsets in general (add it in the same place as `card_eq_one`)

[cjrl]

-awaiting-author

[ghseeli]

-awaiting-author

[b-mehta]

Oops - I should've mentioned this before, but this one should go in Data/Finset/Card, rather than being here. You can also likely avoid some of the variables, because they'll be in the local context already.

[ghseeli]

We went ahead and moved it to the Finset/Card file as you suggested, but we were not quite sure where to put it. It had to go pretty far down in the file, though, since the proof used other lemmas that were down there!