feat: shuffles, permuted lists and other prerequisites for combinatorics

[SashaIr]

This is the first PR in a series of 22 PRs porting the Coq-Combi library to Mathlib.

This PR contains prerequisites for the subsequent port, mainly lemmas about lists, and does not yet include the main Coq-Combi development.

The intended final state of the port is available in my coq-combi branch. The subsequent PRs will progressively introduce the remaining parts of the port.

#mathlib4 > Porting of Coq-Combi @ 💬

---

[github-actions]

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[github-actions]

PR summary ba3d3b0020

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
../mathlib-ci/scripts/pr_summary/import_trans_difference.sh all
There are 6312 files with changed transitive imports taking up over 287577 characters: this is too many to display!
You can run this locally from your mathlib4 directory:

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

---

Declarations diff (regex)

+ Finite.toLattice
+ Finite.toLatticeOfSup
+ IsShuffle
+ IsShuffle.filter_eq_left
+ IsShuffle.filter_eq_right
+ IsShuffle.symm
+ append_mem_shuffle
+ card_permutations_toFinset
+ card_rearrangements
+ card_rearrangements_eq_sum
+ card_rearrangements_mul_prod_factorial
+ countP_eq_card_filter_range
+ eq_of_getD_eq
+ eq_of_length_eq_of_getD_eq
+ exists_bijOn_of_card_fiber_eq
+ exists_equiv_of_card_fiber_eq
+ fiberSubtypeEquiv
+ filterMap_head?_sublist_flatten
+ getD_append_cons_left
+ getD_append_cons_right
+ getD_append_cons_self
+ getD_le_sum
+ getD_tail
+ getD_zero_eq_headD
+ getElem_mid_left
+ getElem_mid_left'
+ getElem_mid_mid
+ getElem_mid_mid'
+ getElem_mid_right
+ getElem_mid_right'
+ getLastD_eq_getD
+ headD_le_sum
+ idxOf_append_singleton
+ isShuffle_append
+ isShuffle_filter
+ isShuffle_nil_left
+ isShuffle_nil_left_iff
+ isShuffle_nil_right
+ isShuffle_nil_right_iff
+ le_getLast_of_pairwise_le
+ length_append_mid
+ length_eq_sum_count
+ length_le_one_of_pairwise
+ length_of_isShuffle
+ length_shuffle
+ mem_flatten_of_mem_getD
+ mem_getD_of_mem_flatten
+ mem_rearrangements
+ mem_shuffle_comm
+ mem_shuffle_iff
+ pairwise_le_getElem?_mono
+ perm_append_of_isShuffle
+ perm_append_of_mem_shuffle
+ prod_factorial_count_erase
+ prod_factorial_count_mul_multinomial
+ rearrangements
+ rearrangements_eq_biUnion
+ rearrangements_zero
+ set_append_cons_left
+ set_append_cons_right
+ set_append_cons_self
+ shuffle
+ shuffle_cons_cons
+ shuffle_nil_left
+ shuffle_nil_right
+ shuffle_perm_comm
+ split_three_getD
+ split_three_getElem
+ sublist_concat_cases
+ sublist_left_of_isShuffle
+ sublist_left_of_mem_shuffle
+ sublist_right_of_isShuffle
+ sublist_right_of_mem_shuffle
+ sublist_three_cases
+ sum_eq_sum_range_getD
+ sum_eq_sum_range_getD_length
+ sum_set_add_getElem
+ sum_tail_add_getD_zero
+ sum_take_eq_sum_range
+ sum_take_le_sum
+ sum_take_succ_getD

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 ba3d3b0).

• +95 new declarations
• −0 removed declarations

+Finite.toLattice
+Finite.toLatticeOfSup
+Fintype.exists_bijOn_of_card_fiber_eq
+Fintype.exists_equiv_of_card_fiber_eq
+Fintype.fiberSubtypeEquiv
+List.IsShuffle
+List.IsShuffle.below
+List.IsShuffle.below.casesOn
+List.IsShuffle.below.left
+List.IsShuffle.below.nil
+List.IsShuffle.below.rec
+List.IsShuffle.below.right
+List.IsShuffle.brecOn
+List.IsShuffle.casesOn
+List.IsShuffle.filter_eq_left
+List.IsShuffle.filter_eq_right
+List.IsShuffle.left
+List.IsShuffle.nil
+List.IsShuffle.rec
+List.IsShuffle.recOn
+List.IsShuffle.right
+List.IsShuffle.symm
+List.append_mem_shuffle
+List.card_permutations_toFinset
+List.card_rearrangements
+List.card_rearrangements_eq_sum
+List.card_rearrangements_mul_prod_factorial
+List.countP_eq_card_filter_range
+List.eq_of_getD_eq
+List.eq_of_length_eq_of_getD_eq
+List.filterMap_head?_sublist_flatten
+List.getD_append_cons_left
+List.getD_append_cons_right
+List.getD_append_cons_self
+List.getD_le_sum
+List.getD_tail
+List.getD_zero_eq_headD
+List.getElem_mid_left
+List.getElem_mid_left'
+List.getElem_mid_mid
+List.getElem_mid_mid'
+List.getElem_mid_right
+List.getElem_mid_right'
+List.getLastD_eq_getD
+List.headD_le_sum
+List.idxOf_append_singleton
+List.isShuffle_append
+List.isShuffle_filter
+List.isShuffle_nil_left
+List.isShuffle_nil_left_iff
+List.isShuffle_nil_right
+List.isShuffle_nil_right_iff
+List.le_getLast_of_pairwise_le
+List.length_append_mid
+List.length_eq_sum_count
+List.length_le_one_of_pairwise
+List.length_of_isShuffle
+List.length_shuffle
+List.mem_flatten_of_mem_getD
+List.mem_getD_of_mem_flatten
+List.mem_rearrangements
+List.mem_shuffle_comm
+List.mem_shuffle_iff
+List.pairwise_le_getElem?_mono
+List.perm_append_of_isShuffle
+List.perm_append_of_mem_shuffle
+List.prod_factorial_count_erase
+List.rearrangements
+List.rearrangements_eq_biUnion
+List.rearrangements_zero
+List.set_append_cons_left
+List.set_append_cons_right
+List.set_append_cons_self
+List.shuffle
+List.shuffle.eq_def
+List.shuffle_cons_cons
+List.shuffle_nil_left
+List.shuffle_nil_right
+List.shuffle_perm_comm
+List.split_three_getD
+List.split_three_getElem
+List.sublist_concat_cases
+List.sublist_left_of_isShuffle
+List.sublist_left_of_mem_shuffle
+List.sublist_right_of_isShuffle
+List.sublist_right_of_mem_shuffle
+List.sublist_three_cases
+List.sum_eq_sum_range_getD
+List.sum_eq_sum_range_getD_length
+List.sum_set_add_getElem
+List.sum_tail_add_getD_zero
+List.sum_take_eq_sum_range
+List.sum_take_le_sum
+List.sum_take_succ_getD
+Multiset.prod_factorial_count_mul_multinomial

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (5.00, 0.00)

Current number	Change	Type (weak)
exposed public sections	5071	5

Current commit ba3d3b0020
Reference commit 8d52ea9a14

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.py 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).

[github-actions]

✅ PR Title Formatted Correctly

The title of this PR has been updated to match our commit style conventions.
Thank you!

[SnirBroshi]

Suggested change

	theorem length_append_mid : (p ++ m ++ s).length = p.length + m.length + s.length := by
	theorem length_append_append : (p ++ m ++ s).length = p.length + m.length + s.length := by

and similarly below

[SnirBroshi]

Suggested change

	(p ++ m ++ s)[i]'(by simp; omega) = p[i] := by
	(p ++ m ++ s)[i]'(by simp; lia) = p[i] := by

or even

Suggested change

	(p ++ m ++ s)[i]'(by simp; omega) = p[i] := by
	(p ++ m ++ s)[i]'(by grind) = p[i] := by

might work. omega is essentially deprecated.

[SnirBroshi]

Suggested change

	theorem filterMap_head?_sublist_flatten (L : List (List α)) :
	(L.filterMap head?).Sublist L.flatten := by
	theorem filterMap_head?_sublist_flatten (L : List (List α)) :
	L.filterMap head? <+ L.flatten := by

[eric-wieser]

As discussed on Zulip, this currently appears to violate the license terms of Coq-combi, as this PR is essentially relicensing derived GPLed code as Apache.

[eric-wieser]

On reflection, I think it is best to close this entirely; otherwise revewers will see this and will have to be careful not to write derived code themselves.