Datasets:
source_repo large_stringclasses 119
values | fact large_stringlengths 6 75.3k | type large_stringclasses 23
values | library large_stringclasses 20
values | imports listlengths 0 34 | filename large_stringclasses 714
values | symbolic_name large_stringlengths 1 134 | docstring stringlengths 2 3.58k ⌀ |
|---|---|---|---|---|---|---|---|
1324-Avoiding-Permutation | G_growth_ratios : list nat :=
map (growth_ratio G_seq) (seq 1 8). | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticAnalysis.G_growth_ratios | null |
1324-Avoiding-Permutation | G_growth_values :
G_growth_ratios = [1000; 2000; 3000; 3833; 4478; 4980; 5384; 5717]%nat.
Proof. vm_compute. reflexivity. Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticAnalysis.G_growth_values | null |
1324-Avoiding-Permutation | estimated_growth : nat :=
extrapolate_limit G_growth_ratios. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticAnalysis.estimated_growth | null |
1324-Avoiding-Permutation | extrapolate_limit (ratios : list nat) : nat :=
let last_few := skipn (length ratios - 3) ratios in
(fold_left Nat.add last_few 0%nat / 3)%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticAnalysis.extrapolate_limit | null |
1324-Avoiding-Permutation | growth_estimate :
(5000 < estimated_growth)%nat /\ (estimated_growth < 6000)%nat.
Proof.
unfold estimated_growth, extrapolate_limit, G_growth_ratios.
vm_compute. lia.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticAnalysis.growth_estimate | null |
1324-Avoiding-Permutation | growth_ratio (seq : list nat) (n : nat) : nat :=
if (nth (n-1) seq 0%nat =? 0)%nat then 0%nat
else (nth n seq 0%nat * 1000 / nth (n-1) seq 0%nat)%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticAnalysis.growth_ratio | null |
1324-Avoiding-Permutation | C (n : nat) : nat := catalan_compute n. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.C | null |
1324-Avoiding-Permutation | R_contribution_increasing :
R_contribution_percent 2 = 50%nat /\
R_contribution_percent 3 = 66%nat /\
R_contribution_percent 4 = 78%nat /\
R_contribution_percent 5 = 86%nat.
Proof.
unfold R_contribution_percent, a, R. repeat split; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.R_contribution_increasing | null |
1324-Avoiding-Permutation | R_contribution_percent (n : nat) : nat :=
if (a n =? 0)%nat then 0%nat
else (R n * 100 / a n)%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.R_contribution_percent | null |
1324-Avoiding-Permutation | R_dominates_from_n4 : forall n, (n >= 4)%nat -> (n <= 5)%nat ->
(R n > C (n - 1))%nat.
Proof.
intros n Hge Hle.
destruct n as [|[|[|[|[|[|]]]]]]; try lia.
- vm_compute. lia.
- vm_compute. lia.
Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.R_dominates_from_n4 | null |
1324-Avoiding-Permutation | R_exceeds_C_at_4 : (R 4 > C 3)%nat.
Proof. unfold R, C. vm_compute. lia. Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.R_exceeds_C_at_4 | null |
1324-Avoiding-Permutation | R_exceeds_C_at_5 : (R 5 > C 4)%nat.
Proof. unfold R, C. vm_compute. lia. Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.R_exceeds_C_at_5 | null |
1324-Avoiding-Permutation | R_exceeds_C_ratio (n : nat) : bool :=
let r_ratio := (R (S n) * 100 / R n)%nat in
let c_ratio := (C (S n) * 100 / C n)%nat in
(c_ratio <? r_ratio)%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.R_exceeds_C_ratio | null |
1324-Avoiding-Permutation | R_grows_faster_than_C :
R_exceeds_C_ratio 2 = true /\
R_exceeds_C_ratio 3 = true /\
R_exceeds_C_ratio 4 = true.
Proof.
unfold R_exceeds_C_ratio, R, C. repeat split; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.R_grows_faster_than_C | null |
1324-Avoiding-Permutation | R_is_dominant_term : forall n, (n >= 4)%nat -> (n <= 5)%nat ->
(R n * 100 / a n > 75)%nat.
Proof.
intros n Hge Hle. unfold R, a.
destruct n as [|[|[|[|[|[|]]]]]]; try lia; vm_compute; lia.
Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.R_is_dominant_term | null |
1324-Avoiding-Permutation | R_to_C_ratio_increasing :
R_to_C_ratio_x100 2 = 100%nat /\
R_to_C_ratio_x100 3 = 200%nat /\
R_to_C_ratio_x100 4 = 360%nat /\
R_to_C_ratio_x100 5 = 635%nat.
Proof.
unfold R_to_C_ratio_x100, R, C. repeat split; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.R_to_C_ratio_increasing | null |
1324-Avoiding-Permutation | R_to_C_ratio_x100 (n : nat) : nat :=
if (n =? 0)%nat then 0%nat
else (R n * 100 / C (n - 1))%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.R_to_C_ratio_x100 | null |
1324-Avoiding-Permutation | a (n : nat) : nat := count_1324_avoiding n. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.a | null |
1324-Avoiding-Permutation | catalan_growth_rate :
(C 1 * 100 / C 0)%nat = 100%nat /\
(C 2 * 100 / C 1)%nat = 200%nat /\
(C 3 * 100 / C 2)%nat = 250%nat /\
(C 4 * 100 / C 3)%nat = 280%nat /\
(C 5 * 100 / C 4)%nat = 300%nat.
Proof.
unfold C. repeat split; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.catalan_growth_rate | null |
1324-Avoiding-Permutation | main_decomposition_verified : forall n, (n >= 1)%nat -> (n <= 5)%nat ->
a n = (C (n - 1) + R n)%nat.
Proof.
intros n Hge Hle. unfold a, C, R.
destruct n as [|[|[|[|[|[|]]]]]]; try lia; vm_compute; reflexivity.
Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | AsymptoticDominance.main_decomposition_verified | null |
1324-Avoiding-Permutation | catalan_compute (n : nat) : nat :=
match n with
| 0%nat => 1%nat
| S n' =>
let fix sum_cat (k : nat) (acc : nat) : nat :=
match k with
| 0%nat => acc
| S k' => sum_cat k' (acc + catalan_compute k' * catalan_compute (n' - k'))%nat
end
in sum_cat n 0%nat
end. | Fixpoint | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CatalanConnection.catalan_compute | null |
1324-Avoiding-Permutation | catalan_gf_coeff (n : nat) : nat := catalan_compute n. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CatalanConnection.catalan_gf_coeff | null |
1324-Avoiding-Permutation | catalan_values :
catalan_compute 0 = 1%nat /\
catalan_compute 1 = 1%nat /\
catalan_compute 2 = 2%nat /\
catalan_compute 3 = 5%nat /\
catalan_compute 4 = 14%nat.
Proof.
repeat split; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CatalanConnection.catalan_values | null |
1324-Avoiding-Permutation | max_at_end_equals_catalan : forall n,
(n >= 1)%nat -> (n <= 5)%nat ->
avoiding_with_max_at_end n = catalan_gf_coeff (n - 1).
Proof.
intros n Hge Hle.
destruct n as [|[|[|[|[|[|]]]]]]; try lia.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
-... | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CatalanConnection.max_at_end_equals_catalan | null |
1324-Avoiding-Permutation | bijection_to_dyck : forall n, (n >= 1)%nat -> (n <= 5)%nat ->
avoiding_with_max_at_end n = count_dyck_paths (n - 1).
Proof. exact complete_chain. Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ComprehensiveSummary.bijection_to_dyck | null |
1324-Avoiding-Permutation | catalan_component : forall n, (n >= 1)%nat -> (n <= 5)%nat ->
avoiding_with_max_at_end n = catalan_compute (n - 1).
Proof. exact max_at_end_equals_catalan. Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ComprehensiveSummary.catalan_component | null |
1324-Avoiding-Permutation | main_theorem_bool : forall sigma n,
(forall x, In x sigma -> (x < n)%nat) ->
avoids_1324 (sigma ++ [n]) = avoids_132 sigma.
Proof. exact catalan_bijection_bool. Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ComprehensiveSummary.main_theorem_bool | null |
1324-Avoiding-Permutation | main_theorem_prop : forall sigma n,
(forall x, In x sigma -> (x < n)%nat) ->
(~ contains_1324 (sigma ++ [n]) <-> ~ contains_132 sigma).
Proof. exact max_end_1324_iff_prefix_132. Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ComprehensiveSummary.main_theorem_prop | null |
1324-Avoiding-Permutation | pattern_containment_chain : forall p,
contains_1324 p -> contains_132 p.
Proof. exact thm_132_subpattern_of_1324. Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ComprehensiveSummary.pattern_containment_chain | null |
1324-Avoiding-Permutation | max_end_1324_iff_prefix_132 : forall prefix n,
(forall x, In x prefix -> (x < n)%nat) ->
(~ contains_1324 (prefix ++ [n]) <-> ~ contains_132 prefix).
Proof.
intros prefix n Hbound.
split.
- apply no_1324_with_max_end_means_no_132_prefix. exact Hbound.
- intros Hno132.
intros H1324.
destruct H1324 as... | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CoreBijectionLemma.max_end_1324_iff_prefix_132 | null |
1324-Avoiding-Permutation | no_1324_with_max_end_means_no_132_prefix : forall prefix n,
(forall x, In x prefix -> (x < n)%nat) ->
~ contains_1324 (prefix ++ [n]) ->
~ contains_132 prefix.
Proof.
intros prefix n Hbound Hno1324 H132.
apply Hno1324.
apply prefix_132_creates_1324.
- exact Hbound.
- exact H132.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CoreBijectionLemma.no_1324_with_max_end_means_no_132_prefix | null |
1324-Avoiding-Permutation | nth_app_left : forall (A : Type) (l1 l2 : list A) (d : A) (i : nat),
(i < length l1)%nat -> nth i (l1 ++ l2) d = nth i l1 d.
Proof.
intros. apply app_nth1. exact H.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CoreBijectionLemma.nth_app_left | null |
1324-Avoiding-Permutation | nth_app_right : forall (A : Type) (l1 l2 : list A) (d : A) (i : nat),
(i >= length l1)%nat -> nth i (l1 ++ l2) d = nth (i - length l1) l2 d.
Proof.
intros. apply app_nth2. lia.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CoreBijectionLemma.nth_app_right | null |
1324-Avoiding-Permutation | prefix_132_creates_1324 : forall prefix n,
(forall x, In x prefix -> (x < n)%nat) ->
contains_132 prefix ->
contains_1324 (prefix ++ [n]).
Proof.
intros prefix n Hmax [i [j [k H132]]].
unfold has_132_at in H132.
destruct H132 as [Hij [Hjk [Hklen [Hvik Hvkj]]]].
exists i, j, k, (length prefix).
unfold ha... | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CoreBijectionLemma.prefix_132_creates_1324 | null |
1324-Avoiding-Permutation | prefix_of_perm_with_max_end (p : list nat) : list nat :=
firstn (length p - 1) p. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CoreBijectionLemma.prefix_of_perm_with_max_end | null |
1324-Avoiding-Permutation | avoiding_with_max_interior (n : nat) : nat :=
let perms := perms_of_n n in
let filtered := filter (fun p => avoids_1324 p && negb (max_at_end p)) perms in
length filtered. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CorrectionTermAnalysis.avoiding_with_max_interior | null |
1324-Avoiding-Permutation | decomposition_complete : forall n, (n <= 5)%nat ->
decomposition_sum n = count_1324_avoiding n.
Proof.
intros n Hle.
unfold decomposition_sum.
destruct n as [|[|[|[|[|[|]]]]]]; try lia.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- vm_com... | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CorrectionTermAnalysis.decomposition_complete | null |
1324-Avoiding-Permutation | decomposition_sum (n : nat) : nat :=
avoiding_with_max_at_end n + avoiding_with_max_interior n. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CorrectionTermAnalysis.decomposition_sum | null |
1324-Avoiding-Permutation | interior_n4 : avoiding_with_max_interior 4 = 18%nat.
Proof. vm_compute. reflexivity. Qed. | Example | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CorrectionTermAnalysis.interior_n4 | null |
1324-Avoiding-Permutation | interior_n5 : avoiding_with_max_interior 5 = 89%nat.
Proof. vm_compute. reflexivity. Qed. | Example | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | CorrectionTermAnalysis.interior_n5 | null |
1324-Avoiding-Permutation | all_perms (l : list nat) : list (list nat) :=
match l with
| [] => [[]]
| x :: xs =>
flat_map (fun p => map (fun i =>
firstn i p ++ [x] ++ skipn i p) (seq 0 (S (length p)))) (all_perms xs)
end. | Fixpoint | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | Counting.all_perms | null |
1324-Avoiding-Permutation | count_1324_avoiding (n : nat) : nat :=
length (filter avoids_1324 (perms_of_n n)). | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | Counting.count_1324_avoiding | null |
1324-Avoiding-Permutation | perms_of_n (n : nat) : list (list nat) :=
all_perms (seq 1 n). | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | Counting.perms_of_n | null |
1324-Avoiding-Permutation | bijection_chain : forall n, (n >= 1)%nat -> (n <= 5)%nat ->
avoiding_with_max_at_end n = count_132_avoiding (n - 1) /\
count_132_avoiding (n - 1) = count_dyck_paths (n - 1).
Proof.
intros n Hge Hle.
destruct n as [|[|[|[|[|[|]]]]]]; try lia; vm_compute; split; reflexivity.
Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.bijection_chain | null |
1324-Avoiding-Permutation | complete_chain : forall n, (n >= 1)%nat -> (n <= 5)%nat ->
avoiding_with_max_at_end n = count_dyck_paths (n - 1).
Proof.
intros n Hge Hle.
destruct n as [|[|[|[|[|[|]]]]]]; try lia; vm_compute; reflexivity.
Qed. | Corollary | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.complete_chain | null |
1324-Avoiding-Permutation | compose_bijection (p : list nat) : dyck_path :=
perm_to_dyck (removelast p). | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.compose_bijection | null |
1324-Avoiding-Permutation | count_dyck_paths (n : nat) : nat :=
match n with
| O => 1%nat
| S n' =>
let fix sum_paths (k : nat) (acc : nat) :=
match k with
| O => acc
| S k' => sum_paths k' (acc + count_dyck_paths k' * count_dyck_paths (n' - k'))%nat
end
in sum_paths n 0%nat
end. | Fixpoint | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.count_dyck_paths | null |
1324-Avoiding-Permutation | dyck_catalan_equivalence : forall n, (n <= 4)%nat ->
count_dyck_paths n = C n.
Proof.
intros n Hle. unfold C.
destruct n as [|[|[|[|[|]]]]]; try lia; vm_compute; reflexivity.
Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.dyck_catalan_equivalence | null |
1324-Avoiding-Permutation | dyck_count_is_catalan :
count_dyck_paths 0 = 1%nat /\
count_dyck_paths 1 = 1%nat /\
count_dyck_paths 2 = 2%nat /\
count_dyck_paths 3 = 5%nat /\
count_dyck_paths 4 = 14%nat.
Proof. repeat split; vm_compute; reflexivity. Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.dyck_count_is_catalan | null |
1324-Avoiding-Permutation | dyck_from_132_avoiding_small :
dyck_path_length (perm_to_dyck [1%nat]) = 2%nat /\
dyck_path_length (perm_to_dyck [1%nat; 2%nat]) = 4%nat /\
dyck_path_length (perm_to_dyck [2%nat; 1%nat]) = 4%nat.
Proof. repeat split; vm_compute; reflexivity. Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.dyck_from_132_avoiding_small | null |
1324-Avoiding-Permutation | dyck_path := list step. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.dyck_path | null |
1324-Avoiding-Permutation | dyck_path_length (d : dyck_path) : nat := length d. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.dyck_path_length | null |
1324-Avoiding-Permutation | is_dyck_path (path : dyck_path) : bool :=
is_valid_prefix path 0 && (length (filter (fun s => match s with Up => true | Down => false end) path) =?
length (filter (fun s => match s with Up => false | Down => true end) path))%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.is_dyck_path | null |
1324-Avoiding-Permutation | is_valid_prefix (path : dyck_path) (height : nat) : bool :=
match path with
| [] => true
| Up :: rest => is_valid_prefix rest (S height)
| Down :: rest =>
match height with
| O => false
| S h => is_valid_prefix rest h
end
end. | Fixpoint | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.is_valid_prefix | null |
1324-Avoiding-Permutation | max_end_1324_to_dyck : forall n, (n >= 1)%nat -> (n <= 5)%nat ->
avoiding_with_max_at_end n = count_dyck_paths (n - 1).
Proof.
intros n Hge Hle.
destruct n as [|[|[|[|[|[|]]]]]]; try lia; vm_compute; reflexivity.
Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.max_end_1324_to_dyck | null |
1324-Avoiding-Permutation | perm_132_to_dyck_preserves_length : forall n, (n <= 4)%nat ->
length (valid_132_avoiding_perms n) = count_dyck_paths n.
Proof.
intros n Hle. unfold valid_132_avoiding_perms.
destruct n as [|[|[|[|[|]]]]]; try lia; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.perm_132_to_dyck_preserves_length | null |
1324-Avoiding-Permutation | perm_to_dyck (p : list nat) : dyck_path :=
perm_to_dyck_aux p []. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.perm_to_dyck | null |
1324-Avoiding-Permutation | perm_to_dyck_aux (p : list nat) (stack : list nat) : dyck_path :=
match p with
| [] => map (fun _ => Down) stack
| x :: xs =>
let downs := length (filter (fun s => (s <? x)%nat) stack) in
let new_stack := filter (fun s => negb (s <? x)%nat) stack in
repeat Down downs ++ [Up] ++ perm_to_dyck_aux ... | Fixpoint | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.perm_to_dyck_aux | null |
1324-Avoiding-Permutation | step : Type :=
| Up : step
| Down : step. | Inductive | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.step | null |
1324-Avoiding-Permutation | valid_132_avoiding_perms (n : nat) : list (list nat) :=
filter avoids_132 (perms_of_n n). | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | DyckPaths.valid_132_avoiding_perms | null |
1324-Avoiding-Permutation | avoiding_132_perms (n : nat) : list (list nat) :=
filter avoids_132 (perms_of_n n). | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ExplicitBijection.avoiding_132_perms | null |
1324-Avoiding-Permutation | bijection_count_match : forall n, (n >= 1)%nat -> (n <= 5)%nat ->
length (max_end_perms n) = length (avoiding_132_perms (n - 1)).
Proof.
intros n Hge Hle.
destruct n as [|[|[|[|[|[|]]]]]]; try lia; vm_compute; reflexivity.
Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ExplicitBijection.bijection_count_match | null |
1324-Avoiding-Permutation | bijection_preserves_avoidance : forall sigma,
Permutation sigma (seq 1 (length sigma)) ->
avoids_1324 (sigma_to_extended sigma) = avoids_132 sigma.
Proof.
intros sigma Hperm.
apply catalan_bijection_bool.
apply sigma_elements_bound.
exact Hperm.
Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ExplicitBijection.bijection_preserves_avoidance | null |
1324-Avoiding-Permutation | extended_sigma_inverse : forall sigma,
extended_to_sigma (sigma_to_extended sigma) = sigma.
Proof.
intros sigma. unfold extended_to_sigma, sigma_to_extended.
apply removelast_app_singleton.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ExplicitBijection.extended_sigma_inverse | null |
1324-Avoiding-Permutation | extended_to_sigma (p : list nat) : list nat :=
removelast p. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ExplicitBijection.extended_to_sigma | null |
1324-Avoiding-Permutation | max_end_perms (n : nat) : list (list nat) :=
filter (fun p => avoids_1324 p && max_at_end p) (perms_of_n n). | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ExplicitBijection.max_end_perms | null |
1324-Avoiding-Permutation | sigma_elements_bound : forall sigma,
Permutation sigma (seq 1 (length sigma)) ->
forall x, In x sigma -> (x < S (length sigma))%nat.
Proof.
intros sigma Hperm x Hin.
apply Permutation_in with (x := x) in Hperm.
- apply in_seq in Hperm. lia.
- exact Hin.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ExplicitBijection.sigma_elements_bound | null |
1324-Avoiding-Permutation | sigma_extended_length : forall sigma,
length (sigma_to_extended sigma) = S (length sigma).
Proof.
intros sigma. unfold sigma_to_extended.
rewrite app_length. simpl. lia.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ExplicitBijection.sigma_extended_length | null |
1324-Avoiding-Permutation | sigma_to_extended (sigma : list nat) : list nat :=
sigma ++ [S (length sigma)]. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | ExplicitBijection.sigma_to_extended | null |
1324-Avoiding-Permutation | bijection_theorem_general :
forall sigma n,
(forall x, In x sigma -> (x < n)%nat) ->
(~ contains_1324 (sigma ++ [n])) <-> (~ contains_132 sigma).
Proof.
exact max_end_1324_iff_prefix_132.
Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FinalSummary.bijection_theorem_general | null |
1324-Avoiding-Permutation | coefficients_relation : forall n, (n <= 5)%nat ->
nth n verified_G_coeffs 0%nat =
(if (n =? 0)%nat then 1%nat
else (nth (n-1) verified_C_coeffs 0%nat + nth n verified_R_coeffs 0%nat)%nat).
Proof.
intros n Hle.
destruct n as [|[|[|[|[|[|]]]]]]; try lia; vm_compute; reflexivity.
Qed. | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FinalSummary.coefficients_relation | null |
1324-Avoiding-Permutation | verified_C_coeffs : list nat := [1; 1; 2; 5; 14; 42]%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FinalSummary.verified_C_coeffs | null |
1324-Avoiding-Permutation | verified_G_coeffs : list nat := [1; 1; 2; 6; 23; 103]%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FinalSummary.verified_G_coeffs | null |
1324-Avoiding-Permutation | verified_R_coeffs : list nat := [0; 0; 1; 4; 18; 89]%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FinalSummary.verified_R_coeffs | null |
1324-Avoiding-Permutation | a_n (n : nat) : nat := count_1324_avoiding n. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationAnalysis.a_n | null |
1324-Avoiding-Permutation | c_n (n : nat) : nat := nth n catalan 0%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationAnalysis.c_n | null |
1324-Avoiding-Permutation | decomposition_formula : forall n, (n >= 1)%nat -> (n <= 5)%nat ->
a_n n = (c_n (n - 1) + r_n n)%nat.
Proof.
intros n Hge Hle.
unfold a_n, c_n, r_n.
destruct n as [|[|[|[|[|[|]]]]]]; try lia.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- v... | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationAnalysis.decomposition_formula | null |
1324-Avoiding-Permutation | interior_n0 : avoiding_with_max_interior 0 = 0%nat.
Proof. vm_compute. reflexivity. Qed. | Example | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationAnalysis.interior_n0 | null |
1324-Avoiding-Permutation | interior_n1 : avoiding_with_max_interior 1 = 0%nat.
Proof. vm_compute. reflexivity. Qed. | Example | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationAnalysis.interior_n1 | null |
1324-Avoiding-Permutation | interior_n2 : avoiding_with_max_interior 2 = 1%nat.
Proof. vm_compute. reflexivity. Qed. | Example | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationAnalysis.interior_n2 | null |
1324-Avoiding-Permutation | interior_n3 : avoiding_with_max_interior 3 = 4%nat.
Proof. vm_compute. reflexivity. Qed. | Example | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationAnalysis.interior_n3 | null |
1324-Avoiding-Permutation | interior_sequence : list nat := [0; 0; 1; 4; 18; 89]%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationAnalysis.interior_sequence | null |
1324-Avoiding-Permutation | interior_values : forall n, (n <= 5)%nat ->
avoiding_with_max_interior n = nth n interior_sequence 0%nat.
Proof.
intros n Hle.
destruct n as [|[|[|[|[|[|]]]]]]; try lia.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- vm_compute. reflexivity.
- vm_compute. reflexivity.... | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationAnalysis.interior_values | null |
1324-Avoiding-Permutation | r_n (n : nat) : nat := avoiding_with_max_interior n. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationAnalysis.r_n | null |
1324-Avoiding-Permutation | C_squared_conv (n : nat) : nat :=
convolution catalan_compute catalan_compute n. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationRefinement.C_squared_conv | null |
1324-Avoiding-Permutation | C_squared_values :
C_squared_conv 0 = 1%nat /\
C_squared_conv 1 = 2%nat /\
C_squared_conv 2 = 5%nat /\
C_squared_conv 3 = 14%nat /\
C_squared_conv 4 = 42%nat.
Proof.
repeat split; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationRefinement.C_squared_values | null |
1324-Avoiding-Permutation | convolution (f g : nat -> nat) (n : nat) : nat :=
fold_left Nat.add (map (fun k => f k * g (n - k))%nat (seq 0 (S n))) 0%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationRefinement.convolution | null |
1324-Avoiding-Permutation | functional_eq_matches_G : forall n,
(n <= 5)%nat ->
functional_eq_rhs n = nth n G_seq 0%nat.
Proof.
intros n Hle.
unfold functional_eq_rhs, G_seq, R_seq_extended.
destruct n as [|[|[|[|[|[|]]]]]]; try lia; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationRefinement.functional_eq_matches_G | null |
1324-Avoiding-Permutation | functional_eq_rhs (n : nat) : nat :=
if (n =? 0)%nat then 1%nat
else if (n =? 1)%nat then 1%nat
else (catalan_compute (n - 1) + nth n R_seq_extended 0%nat)%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationRefinement.functional_eq_rhs | null |
1324-Avoiding-Permutation | geometric_C (n : nat) : nat :=
fold_left Nat.add (map catalan_compute (seq 0 (S n))) 0%nat. | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationRefinement.geometric_C | null |
1324-Avoiding-Permutation | geometric_C_values :
geometric_C 0 = 1%nat /\
geometric_C 1 = 2%nat /\
geometric_C 2 = 4%nat /\
geometric_C 3 = 9%nat /\
geometric_C 4 = 23%nat.
Proof.
repeat split; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FunctionalEquationRefinement.geometric_C_values | null |
1324-Avoiding-Permutation | left_avoids_132_count (n : nat) : nat :=
let perms := perms_of_n n in
length (filter (fun p =>
avoids_1324 p &&
negb (max_at_end p) &&
avoids_132 (left_of_max p)
) perms). | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FurtherStructure.left_avoids_132_count | null |
1324-Avoiding-Permutation | left_avoids_132_values :
left_avoids_132_count 3 = 4%nat /\
left_avoids_132_count 4 = 18%nat.
Proof.
repeat split; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FurtherStructure.left_avoids_132_values | null |
1324-Avoiding-Permutation | right_avoids_132_count (n : nat) : nat :=
let perms := perms_of_n n in
length (filter (fun p =>
avoids_1324 p &&
negb (max_at_end p) &&
avoids_132 (right_of_max p)
) perms). | Definition | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FurtherStructure.right_avoids_132_count | null |
1324-Avoiding-Permutation | right_avoids_132_values :
right_avoids_132_count 3 = 4%nat /\
right_avoids_132_count 4 = 17%nat.
Proof.
repeat split; vm_compute; reflexivity.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | FurtherStructure.right_avoids_132_values | null |
1324-Avoiding-Permutation | append_singleton_injective : forall (A : Type) (l1 l2 : list A) (x : A),
l1 ++ [x] = l2 ++ [x] -> l1 = l2.
Proof.
intros A l1 l2 x Heq.
apply (f_equal (@removelast A)) in Heq.
rewrite !removelast_app_singleton in Heq.
exact Heq.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | GeneralCatalanBijection.append_singleton_injective | null |
1324-Avoiding-Permutation | append_singleton_length : forall (A : Type) (l : list A) (x : A),
length (l ++ [x]) = S (length l).
Proof.
intros. rewrite app_length. simpl. lia.
Qed. | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | GeneralCatalanBijection.append_singleton_length | null |
1324-Avoiding-Permutation | avoids_1324_iff_not_contains : forall p,
avoids_1324 p = true <-> ~ contains_1324 p.
Proof.
intros p.
unfold avoids_1324, contains_1324_subseq, contains_1324.
rewrite negb_true_iff.
rewrite existsb_false_forall.
split.
- intros Hav [i [j [k [l H]]]].
unfold has_1324_at in H.
destruct H as [Hij [Hj... | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | GeneralCatalanBijection.avoids_1324_iff_not_contains | null |
1324-Avoiding-Permutation | avoids_132_iff_not_contains : forall p,
avoids_132 p = true <-> ~ contains_132 p.
Proof.
intros p.
unfold avoids_132, contains_132_subseq, contains_132.
rewrite negb_true_iff.
rewrite existsb_false_forall.
split.
- intros Hav [i [j [k H]]].
unfold has_132_at in H.
destruct H as [Hij [Hjk [Hklen [H... | Lemma | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | GeneralCatalanBijection.avoids_132_iff_not_contains | null |
1324-Avoiding-Permutation | catalan_bijection_bool : forall prefix n,
(forall x, In x prefix -> (x < n)%nat) ->
avoids_1324 (prefix ++ [n]) = avoids_132 prefix.
Proof.
intros prefix n Hbound.
destruct (avoids_132 prefix) eqn:E132.
- apply avoids_1324_iff_not_contains.
apply avoids_132_iff_not_contains in E132.
apply max_end_1324... | Theorem | Root | [
"Require Import Coq.Lists.List.",
"Require Import Coq.Arith.Arith.",
"Require Import Coq.Bool.Bool.",
"Require Import Coq.Sorting.Permutation.",
"Require Import Lia.",
"Require Import ZArith."
] | Avoid1324.v | GeneralCatalanBijection.catalan_bijection_bool | null |
End of preview. Expand in Data Studio
Omnia
Cross-domain Coq declarations from the Omnia
formalization corpus, plus several library projects. One unified dataset combining
declarations from every backed-up source repo with .v files.
The corpus spans law, medicine, engineering, history, mathematics, networking, cultural systems, and more. Every entry is a Coq declaration extracted from a machine-checked source repository.
Part of the phanerozoic proof assistant projects collection.
Schema
One row per declaration:
| column | type | description |
|---|---|---|
source_repo |
string | GitHub repo name (e.g. acls-verified, CoqForge) |
fact |
string | declaration body, NOT including the leading keyword |
type |
string | declaration kind (Definition, Lemma, Theorem, Inductive, Record, etc.) |
library |
string | sub-library (subdirectory of theories/ or repo root) |
imports |
list[string] | Require Import statements active for the file |
filename |
string | source .v file path relative to the repo root |
symbolic_name |
string | declaration identifier, qualified by enclosing Module/Section |
docstring |
string | preceding (** ... *) doc comment, nullable |
Rows are sorted by (source_repo, filename, symbolic_name) for stable diffs
across regenerations.
Statistics
- Total declarations: 71,021
- Source repos contributing rows: 119
- Source
.vfiles processed: 715 - Sub-libraries: 20
- Declarations with docstrings: 19,118 (27%)
- Top declaration types: Definition: 30453, Lemma: 20160, Theorem: 7893, Record: 3626, Example: 2688, Inductive: 2563, Fixpoint: 1438, Variable: 376
- Top contributing repos: CoqForge (6796), Sammath-Naur (5216), AI-Bootstrap (4629), proof2weights (4319), antikythera-verified (3527), acls-verified (2252), apgar-verified (1872), physics-verified (1497), rfna-verified (1484), transfusion-verified (1393)
Coverage
Domains include but are not limited to:
- Legal/Regulatory: tax law (1031 exchanges), maritime law (UNCLOS), constitutional structures, ancient legal codes
- Medical/Clinical: ACLS protocols, APGAR scoring, sepsis criteria, transfusion compatibility, anesthesia
- Engineering/Technical: avionics standards (ARINC 429), networking RFCs, spacecraft systems, electrical codes
- Historical: battle chronologies, ancient civilizations, calendar systems
- Mathematical: combinatorics, category theory, number theory, type theory
- Cultural: chess variants, musical theory, liturgical calendars
- Software: connected component labeling, theorem proving infrastructure
Extraction
Each row is produced by a deterministic Python extractor that:
- Tokenises Coq source while respecting nested comments, doc comments, and strings.
- Identifies top-level declarations by keyword.
- Determines each declaration's terminating sentence (
.followed by whitespace) or proof terminator (Qed./Defined./Admitted./Abort./Save.), based on whether a proof block immediately follows the declaration head. - Captures the immediately preceding
(** ... *)doc comment when present. - Tracks
Module/Sectionnesting to qualify declaration names.
Same source state always produces the same parquet (rows in deterministic order).
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