feat: extensible push_neg tactic

[JovanGerb]

This was started by point (3) of https://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/Possible.20improvements.20to.20Lean.2FMathlib/near/499085779

To be able to use push_neg more generally, this PR reimplements the tactic so that it is extensible. So, users can tag lemmas to be used by push_neg.

This PR tags lemmas concerning empty/nonempty, subsingleton/nontrivial, and finite/infinite to be used in push_neg.

Additionally, the tactic is extended to be able to push any constant that you want to push, and push_neg is now a macro for push Not. So far, I've tried to add lemmas for pushing Real.log. I could imagine it could also be used for Real.exp, pow, mul, smul, or even Set.mem. (the Set.mem lemmas are mostly tagged with @[simp] already, but push_mem could be more readable than simp or dsimp). Or a combination of pushing Not and Set.compl could be useful.

The implementation is with the simp infrastructure, but instead of using post lemmas, like in simp, push uses pre lemmas, so we rewrite each expression before its subterms are rewritten. This is a more efficient way to "push" constants recursively. Most lemmas are added via the @[push] attribute, but a few cases require some special handling, so they are implemented "by hand" in a simproc that is tried after the push lemmas.

For example the simproc simplifies ¬∀ n, p ⇨ ∃ n, ¬p, but, this is overridden by the push lemma that rewrites ¬(p → q) ⇨ p ∧ ¬q (which doesn't apply at every forall)

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[github-actions]

PR summary a7bfbd40d3

Import changes exceeding 2%

%	File
+3.48%	Mathlib.Algebra.Group.Pi.Basic
+5.56%	Mathlib.Logic.IsEmpty
+2.70%	Mathlib.Logic.Nontrivial.Defs
+2.22%	Mathlib.Order.Defs.LinearOrder

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Tactic.PushNeg	62	0	-62 (-100.00%)
Mathlib.Tactic.Contrapose	63	45	-18 (-28.57%)
Mathlib.Tactic.ByContra	64	46	-18 (-28.12%)
Mathlib.Logic.IsEmpty	72	76	+4 (+5.56%)
Mathlib.Algebra.Group.Pi.Basic	115	119	+4 (+3.48%)
Mathlib.Tactic.Common	241	239	-2 (-0.83%)

Import changes for all files

Files	Import difference
There are 5500 files with changed transitive imports taking up over 232079 characters: this is too many to display!
You can run scripts/import_trans_difference.sh all locally to see the whole output.

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Declarations diff

+ Head
+ Head.isHeadOf
+ Nat.not_nonneg_iff_eq_zero
+ PushSimpConfig
+ elabHead
+ elabPush
+ elabPushConv
+ mem_compl
+ nonempty_iff_empty_ne'
+ not_finite
+ not_imp
+ not_not
+ not_one_lt_iff_eq_one
+ not_zero_lt_iff
+ pushCore
+ pushStep
++ nonempty_iff_empty_ne
+-+ not_iff
- elabPushNegConv
- empty_ne_eq_nonempty
- ne_empty_eq_nonempty
- not_ge_eq
- not_gt_eq
- not_implies_eq
- not_le_eq
- not_lt_eq
- not_ne_eq
- not_nonempty_eq
- not_not_eq
- not_or_eq
- pushNegCore
- transformNegationStep

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.

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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).

[grunweg]

Thanks a lot for starting this discussion! I for one believe a push_neg attribute would be rather nice (but I don't know how hard this would make it to write this tactic).