feat(SetTheory/Cardinal): strong induction on Nat.card for finite types

[rosborn]

Add the Finite analogue for induction_subsingleton_or_nontrivial.

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This version uses a bare motive Type* → Prop since Finite carries no data. This allows it to be used directly with the induction tactic. I've found I've needed this when working with finite groups. It also cleans up the current IsPGroup.isNilpotent proof.

[github-actions]

PR summary 8141c6231d

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff (regex)

+ induction_subsingleton_or_nontrivial

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 8141c62).

• +1 new declarations
• −0 removed declarations

+Finite.induction_subsingleton_or_nontrivial

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No changes to strong technical debt.

No changes to weak technical debt.

Current commit 8141c6231d
Reference commit 910a7566b3

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).