From VLSM.Lib Require Import Itauto.[Loading ML file ring_plugin.cmxs (using legacy method) ... done]
[Loading ML file zify_plugin.cmxs (using legacy method) ... done]
[Loading ML file micromega_plugin.cmxs (using legacy method) ... done]
[Loading ML file btauto_plugin.cmxs (using legacy method) ... done]
[Loading ML file coq-itauto.plugin ... done]
From stdpp Require Import prelude.
From VLSM.Lib Require Import Preamble ListExtras.

Utility: Std++ General Results

Lemma elem_of_take {A : Type} (l : list A) (n : nat) (x : A) :
  elem_of x (take n l) -> elem_of x l.A: Type
l: list A
n: nat
x: A
x ∈ take n l → x ∈ l
Proof.A: Type
l: list A
n: nat
x: A
x ∈ take n l → x ∈ l
  generalize dependent n.A: Type
l: list A
x: A
∀ n : nat, x ∈ take n l → x ∈ l
  induction l; intros n H.A: Type
x: A
n: nat
H: x ∈ take n []
x ∈ []
A: Type
a: A
l: list A
x: A
IHl: ∀ n : nat, x ∈ take n l → x ∈ l
n: nat
H: x ∈ take n (a :: l)
x ∈ a :: l
  -A: Type
x: A
n: nat
H: x ∈ take n []
x ∈ [] by simpl in H; destruct n; simpl in H; inversion H.
  -A: Type
a: A
l: list A
x: A
IHl: ∀ n : nat, x ∈ take n l → x ∈ l
n: nat
H: x ∈ take n (a :: l)
x ∈ a :: l destruct n; [by inversion H |].A: Type
a: A
l: list A
x: A
IHl: ∀ n : nat, x ∈ take n l → x ∈ l
n: nat
H: x ∈ take (S n) (a :: l)
x ∈ a :: l
    simpl in H.A: Type
a: A
l: list A
x: A
IHl: ∀ n : nat, x ∈ take n l → x ∈ l
n: nat
H: x ∈ a :: take n l
x ∈ a :: l
    apply elem_of_cons in H as [-> | H].A: Type
a: A
l: list A
IHl: ∀ n : nat, a ∈ take n l → a ∈ l
n: nat
a ∈ a :: l
A: Type
a: A
l: list A
x: A
IHl: ∀ n : nat, x ∈ take n l → x ∈ l
n: nat
H: x ∈ take n l
x ∈ a :: l
    +A: Type
a: A
l: list A
IHl: ∀ n : nat, a ∈ take n l → a ∈ l
n: nat
a ∈ a :: l by left.
    +A: Type
a: A
l: list A
x: A
IHl: ∀ n : nat, x ∈ take n l → x ∈ l
n: nat
H: x ∈ take n l
x ∈ a :: l by right; eapply IHl.
Qed.

Lemma map_tail [A B : Type] (f : A -> B) (l : list A) :
  map f (tail l) = tail (map f l).A, B: Type
f: A → B
l: list A
map f (tail l) = tail (map f l)
Proof.A, B: Type
f: A → B
l: list A
map f (tail l) = tail (map f l)
  by destruct l.
Qed.

Lemma nth_error_stdpp_last {A : Type} (l : list A) :
  nth_error l (length l - 1) = last l.A: Type
l: list A
nth_error l (length l - 1) = last l
Proof.A: Type
l: list A
nth_error l (length l - 1) = last l
  induction l; [done |].A: Type
a: A
l: list A
IHl: nth_error l (length l - 1) = last l
nth_error (a :: l) (length (a :: l) - 1) =
last (a :: l)
  destruct l; [done |]; cbn in *.A: Type
a, a0: A
l: list A
IHl: nth_error (a0 :: l) (length l - 0) = last (a0 :: l)
nth_error (a0 :: l) (length l) = last (a0 :: l)
  by rewrite <- IHl, Nat.sub_0_r.
Qed.

Lemma last_last_error {A : Type} (l : list A) :
  last_error l = last l.A: Type
l: list A
last_error l = last l
Proof.A: Type
l: list A
last_error l = last l
  induction l; [done |].A: Type
a: A
l: list A
IHl: last_error l = last l
last_error (a :: l) = last (a :: l)
  rewrite last_cons, <- IHl; clear IHl.A: Type
a: A
l: list A
last_error (a :: l) =
match last_error l with
| Some y => Some y
| None => Some a
end
  destruct l; [done |]; cbn; f_equal.A: Type
a, a0: A
l: list A
match l with
| [] => a0
| _ :: _ => List.last l a
end = List.last l a0
  induction l; [done |]; cbn.A: Type
a, a0, a1: A
l: list A
IHl: match l with
| [] => a0
| _ :: _ => List.last l a
end = List.last l a0
match l with
| [] => a1
| _ :: _ => List.last l a
end =
match l with
| [] => a1
| _ :: _ => List.last l a0
end
  rewrite <- IHl.A: Type
a, a0, a1: A
l: list A
IHl: match l with
| [] => a0
| _ :: _ => List.last l a
end = List.last l a0
match l with
| [] => a1
| _ :: _ => List.last l a
end =
match l with
| [] => a1
| _ :: _ =>
    match l with
    | [] => a0
    | _ :: _ => List.last l a
    end
end
  by destruct l.
Qed.

Lemma existsb_Exists {A} (f : A -> bool) :
  forall l, existsb f l = true <-> Exists (fun x => f x = true) l.A: Type
f: A → bool
∀ l : list A,
  existsb f l = true ↔ Exists (λ x : A, f x = true) l
Proof.A: Type
f: A → bool
∀ l : list A,
  existsb f l = true ↔ Exists (λ x : A, f x = true) l
  intro l.A: Type
f: A → bool
l: list A
existsb f l = true ↔ Exists (λ x : A, f x = true) l
  rewrite Exists_exists, existsb_exists.A: Type
f: A → bool
l: list A
(∃ x : A, In x l ∧ f x = true)
↔ (∃ x : A, x ∈ l ∧ f x = true)
  apply exist_proper; intros x.A: Type
f: A → bool
l: list A
x: A
In x l ∧ f x = true ↔ x ∈ l ∧ f x = true
  by rewrite elem_of_list_In.
Qed.

Lemma Exists_last
  {A : Type}
  (l : list A)
  (P : A -> Prop)
  (Pdec : forall a, Decision (P a))
  (Hsomething : Exists P l)
  : exists (prefix : list A)
         (suffix : list A)
         (last : A),
         P last /\
         l = prefix ++ [last] ++ suffix /\
         ~ Exists P suffix.A: Type
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
Hsomething: Exists P l
∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l = prefix ++ [last] ++ suffix ∧ ¬ Exists P suffix

Proof.A: Type
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
Hsomething: Exists P l
∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l = prefix ++ [last] ++ suffix ∧ ¬ Exists P suffix
  induction l using rev_ind; [by inversion Hsomething |].A: Type
x: A
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
Hsomething: Exists P (l ++ [x])
IHl: Exists P l
→ ∃ (prefix suffix : list A) (last : A),
    P last
    ∧ l = prefix ++ [last] ++ suffix
      ∧ ¬ Exists P suffix
∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l ++ [x] = prefix ++ [last] ++ suffix
    ∧ ¬ Exists P suffix
  destruct (decide (P x)).A: Type
x: A
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
Hsomething: Exists P (l ++ [x])
IHl: Exists P l
→ ∃ (prefix suffix : list A) (last : A),
    P last
    ∧ l = prefix ++ [last] ++ suffix
      ∧ ¬ Exists P suffix
p: P x
∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l ++ [x] = prefix ++ [last] ++ suffix
    ∧ ¬ Exists P suffix
A: Type
x: A
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
Hsomething: Exists P (l ++ [x])
IHl: Exists P l
→ ∃ (prefix suffix : list A) (last : A),
    P last
    ∧ l = prefix ++ [last] ++ suffix
      ∧ ¬ Exists P suffix
n: ¬ P x
∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l ++ [x] = prefix ++ [last] ++ suffix
    ∧ ¬ Exists P suffix
  -A: Type
x: A
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
Hsomething: Exists P (l ++ [x])
IHl: Exists P l
→ ∃ (prefix suffix : list A) (last : A),
    P last
    ∧ l = prefix ++ [last] ++ suffix
      ∧ ¬ Exists P suffix
p: P x
∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l ++ [x] = prefix ++ [last] ++ suffix
    ∧ ¬ Exists P suffix exists l, nil, x.A: Type
x: A
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
Hsomething: Exists P (l ++ [x])
IHl: Exists P l
→ ∃ (prefix suffix : list A) (last : A),
    P last
    ∧ l = prefix ++ [last] ++ suffix
      ∧ ¬ Exists P suffix
p: P x
P x ∧ l ++ [x] = l ++ [x] ++ [] ∧ ¬ Exists P []
    rewrite Exists_nil.A: Type
x: A
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
Hsomething: Exists P (l ++ [x])
IHl: Exists P l
→ ∃ (prefix suffix : list A) (last : A),
    P last
    ∧ l = prefix ++ [last] ++ suffix
      ∧ ¬ Exists P suffix
p: P x
P x ∧ l ++ [x] = l ++ [x] ++ [] ∧ ¬ False
    by itauto.
  -A: Type
x: A
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
Hsomething: Exists P (l ++ [x])
IHl: Exists P l
→ ∃ (prefix suffix : list A) (last : A),
    P last
    ∧ l = prefix ++ [last] ++ suffix
      ∧ ¬ Exists P suffix
n: ¬ P x
∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l ++ [x] = prefix ++ [last] ++ suffix
    ∧ ¬ Exists P suffix apply Exists_app in Hsomething.A: Type
x: A
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
Hsomething: Exists P l ∨ Exists P [x]
IHl: Exists P l
→ ∃ (prefix suffix : list A) (last : A),
    P last
    ∧ l = prefix ++ [last] ++ suffix
      ∧ ¬ Exists P suffix
n: ¬ P x
∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l ++ [x] = prefix ++ [last] ++ suffix
    ∧ ¬ Exists P suffix
    destruct Hsomething; [| by inversion H; [| inversion H1]].A: Type
x: A
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
H: Exists P l
IHl: Exists P l
→ ∃ (prefix suffix : list A) (last : A),
    P last
    ∧ l = prefix ++ [last] ++ suffix
      ∧ ¬ Exists P suffix
n: ¬ P x
∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l ++ [x] = prefix ++ [last] ++ suffix
    ∧ ¬ Exists P suffix
    specialize (IHl H); clear H.A: Type
x: A
l: list A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
IHl: ∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l = prefix ++ [last] ++ suffix
    ∧ ¬ Exists P suffix
n: ¬ P x
∃ (prefix suffix : list A) (last : A),
  P last
  ∧ l ++ [x] = prefix ++ [last] ++ suffix
    ∧ ¬ Exists P suffix
    destruct IHl as [prefix [suffix [last [Hf [-> Hnone_after]]]]].A: Type
x: A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
prefix, suffix: list A
last: A
Hf: P last
Hnone_after: ¬ Exists P suffix
n: ¬ P x
∃ (prefix0 suffix0 : list A) (last0 : A),
  P last0
  ∧ (prefix ++ [last] ++ suffix) ++ [x] =
    prefix0 ++ [last0] ++ suffix0 ∧ ¬ Exists P suffix0
    exists prefix, (suffix ++ [x]), last.A: Type
x: A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
prefix, suffix: list A
last: A
Hf: P last
Hnone_after: ¬ Exists P suffix
n: ¬ P x
P last
∧ (prefix ++ [last] ++ suffix) ++ [x] =
  prefix ++ [last] ++ suffix ++ [x]
  ∧ ¬ Exists P (suffix ++ [x])
    simpl.A: Type
x: A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
prefix, suffix: list A
last: A
Hf: P last
Hnone_after: ¬ Exists P suffix
n: ¬ P x
P last
∧ (prefix ++ last :: suffix) ++ [x] =
  prefix ++ last :: suffix ++ [x]
  ∧ ¬ Exists P (suffix ++ [x]) rewrite <- app_assoc.A: Type
x: A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
prefix, suffix: list A
last: A
Hf: P last
Hnone_after: ¬ Exists P suffix
n: ¬ P x
P last
∧ prefix ++ (last :: suffix) ++ [x] =
  prefix ++ last :: suffix ++ [x]
  ∧ ¬ Exists P (suffix ++ [x]) simpl.A: Type
x: A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
prefix, suffix: list A
last: A
Hf: P last
Hnone_after: ¬ Exists P suffix
n: ¬ P x
P last
∧ prefix ++ last :: suffix ++ [x] =
  prefix ++ last :: suffix ++ [x]
  ∧ ¬ Exists P (suffix ++ [x])
    rewrite Exists_app.A: Type
x: A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
prefix, suffix: list A
last: A
Hf: P last
Hnone_after: ¬ Exists P suffix
n: ¬ P x
P last
∧ prefix ++ last :: suffix ++ [x] =
  prefix ++ last :: suffix ++ [x]
  ∧ ¬ (Exists P suffix ∨ Exists P [x]) rewrite Exists_cons.A: Type
x: A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
prefix, suffix: list A
last: A
Hf: P last
Hnone_after: ¬ Exists P suffix
n: ¬ P x
P last
∧ prefix ++ last :: suffix ++ [x] =
  prefix ++ last :: suffix ++ [x]
  ∧ ¬ (Exists P suffix ∨ P x ∨ Exists P []) rewrite Exists_nil.A: Type
x: A
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
prefix, suffix: list A
last: A
Hf: P last
Hnone_after: ¬ Exists P suffix
n: ¬ P x
P last
∧ prefix ++ last :: suffix ++ [x] =
  prefix ++ last :: suffix ++ [x]
  ∧ ¬ (Exists P suffix ∨ P x ∨ False)
    by itauto.
Qed.

Lemma existsb_last
  {A : Type}
  (l : list A)
  (f : A -> bool)
  (Hsomething : existsb f l = true) :
  exists (prefix : list A)
         (suffix : list A)
         (last : A),
         (f last = true) /\
         l = prefix ++ [last] ++ suffix /\
         (existsb f suffix = false).A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∃ (prefix suffix : list A) (last : A),
  f last = true
  ∧ l = prefix ++ [last] ++ suffix
    ∧ existsb f suffix = false
Proof.A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∃ (prefix suffix : list A) (last : A),
  f last = true
  ∧ l = prefix ++ [last] ++ suffix
    ∧ existsb f suffix = false
  setoid_rewrite <-not_true_iff_false.A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∃ (prefix suffix : list A) (last : A),
  f last = true
  ∧ l = prefix ++ [last] ++ suffix
    ∧ existsb f suffix ≠ true
  setoid_rewrite existsb_Exists.A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∃ (prefix suffix : list A) (last : A),
  f last = true
  ∧ l = prefix ++ [last] ++ suffix
    ∧ ¬ Exists (λ x : A, f x = true) suffix
  apply Exists_last.A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∀ a : A, Decision (f a = true)
A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
Exists (λ x : A, f x = true) l
  -A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∀ a : A, Decision (f a = true) by typeclasses eauto.
  -A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
Exists (λ x : A, f x = true) l by apply existsb_Exists.
Qed.

Lemma existsb_forall {A} (f : A -> bool) :
  forall l, existsb f l = false <-> forall x, x ∈ l -> f x = false.A: Type
f: A → bool
∀ l : list A,
  existsb f l = false ↔ (∀ x : A, x ∈ l → f x = false)
Proof.A: Type
f: A → bool
∀ l : list A,
  existsb f l = false ↔ (∀ x : A, x ∈ l → f x = false)
  intro l.A: Type
f: A → bool
l: list A
existsb f l = false ↔ (∀ x : A, x ∈ l → f x = false)
  setoid_rewrite <- not_true_iff_false.A: Type
f: A → bool
l: list A
existsb f l ≠ true ↔ (∀ x : A, x ∈ l → f x ≠ true)
  by rewrite existsb_Exists, <- Forall_Exists_neg, Forall_forall.
Qed.

Lemma existsb_first
  {A : Type}
  (l : list A)
  (f : A -> bool)
  (Hsomething : existsb f l = true) :
  exists (prefix : list A)
         (suffix : list A)
         (first : A),
         (f first = true) /\
         l = prefix ++ [first] ++ suffix /\
         (existsb f prefix = false).A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∃ (prefix suffix : list A) (first : A),
  f first = true
  ∧ l = prefix ++ [first] ++ suffix
    ∧ existsb f prefix = false
Proof.A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∃ (prefix suffix : list A) (first : A),
  f first = true
  ∧ l = prefix ++ [first] ++ suffix
    ∧ existsb f prefix = false
  setoid_rewrite <-not_true_iff_false.A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∃ (prefix suffix : list A) (first : A),
  f first = true
  ∧ l = prefix ++ [first] ++ suffix
    ∧ existsb f prefix ≠ true
  setoid_rewrite existsb_Exists.A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∃ (prefix suffix : list A) (first : A),
  f first = true
  ∧ l = prefix ++ [first] ++ suffix
    ∧ ¬ Exists (λ x : A, f x = true) prefix
  apply Exists_first.A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∀ a : A, Decision (f a = true)
A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
Exists (λ x : A, f x = true) l
  -A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
∀ a : A, Decision (f a = true) by typeclasses eauto.
  -A: Type
l: list A
f: A → bool
Hsomething: existsb f l = true
Exists (λ x : A, f x = true) l by apply existsb_Exists.
Qed.

(*
  Returns all elements <<X>> of <<l>> such that <<X>> does not compare less
  than any other element w.r.t to the precedes relation.
*)
Definition maximal_elements_list
  {A} (precedes : relation A) `{!RelDecision precedes} (l : list A)
  : list A :=
  filter (fun a => Forall (fun b => ~ precedes a b) l) l.

Example maximal_elements_list1 : maximal_elements_list Nat.lt [1; 4; 2; 4] = [4; 4].maximal_elements_list Nat.lt [1; 4; 2; 4] = [4; 4]
Proof.maximal_elements_list Nat.lt [1; 4; 2; 4] = [4; 4] by itauto. Qed.

Example maximal_elements_list2 : maximal_elements_list Nat.le [1; 4; 2; 4] = [].maximal_elements_list Nat.le [1; 4; 2; 4] = []
Proof.maximal_elements_list Nat.le [1; 4; 2; 4] = [] by itauto. Qed.

Returns all elements x of a set S such that x does not compare less than any other element in S w.r.t to a given precedes relation.

Definition maximal_elements_set
  `{HfinSetMessage : FinSet A SetA}
   (precedes : relation A) `{!RelDecision precedes} (s : SetA)
   : SetA :=
    filter (fun a => set_Forall (fun b => ~ precedes a b) s) s.

Lemma filter_ext_elem_of {A} P Q
 `{forall (x : A), Decision (P x)} `{forall (x : A), Decision (Q x)} (l : list A) :
 (forall a, a ∈ l -> (P a <-> Q a)) ->
 filter P l = filter Q l.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
l: list A
(∀ a : A, a ∈ l → P a ↔ Q a) → filter P l = filter Q l
Proof.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
l: list A
(∀ a : A, a ∈ l → P a ↔ Q a) → filter P l = filter Q l
  induction l; intros.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, a ∈ [] → P a ↔ Q a
filter P [] = filter Q []
A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
a: A
l: list A
IHl: (∀ a : A, a ∈ l → P a ↔ Q a) → filter P l = filter Q l
H1: ∀ a0 : A, a0 ∈ a :: l → P a0 ↔ Q a0
filter P (a :: l) = filter Q (a :: l)
  -A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, a ∈ [] → P a ↔ Q a
filter P [] = filter Q [] by rewrite 2 filter_nil.
  -A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
a: A
l: list A
IHl: (∀ a : A, a ∈ l → P a ↔ Q a) → filter P l = filter Q l
H1: ∀ a0 : A, a0 ∈ a :: l → P a0 ↔ Q a0
filter P (a :: l) = filter Q (a :: l) rewrite 2 filter_cons.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
a: A
l: list A
IHl: (∀ a : A, a ∈ l → P a ↔ Q a) → filter P l = filter Q l
H1: ∀ a0 : A, a0 ∈ a :: l → P a0 ↔ Q a0
(if decide (P a) then a :: filter P l else filter P l) =
(if decide (Q a) then a :: filter Q l else filter Q l)
    setoid_rewrite elem_of_cons in H1.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
a: A
l: list A
IHl: (∀ a : A, a ∈ l → P a ↔ Q a) → filter P l = filter Q l
H1: ∀ a0 : A, a0 = a ∨ a0 ∈ l → P a0 ↔ Q a0
(if decide (P a) then a :: filter P l else filter P l) =
(if decide (Q a) then a :: filter Q l else filter Q l)
    by destruct (decide (P a)), (decide (Q a)); [rewrite IHl | ..]; firstorder.
Qed.

Lemma ext_elem_of_filter {A} P Q
 `{forall (x : A), Decision (P x)} `{forall (x : A), Decision (Q x)}
 (l : list A) :
 filter P l = filter Q l -> forall a, a ∈ l -> (P a <-> Q a).A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
l: list A
filter P l = filter Q l → ∀ a : A, a ∈ l → P a ↔ Q a
Proof.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
l: list A
filter P l = filter Q l → ∀ a : A, a ∈ l → P a ↔ Q a
  intros.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
l: list A
H1: filter P l = filter Q l
a: A
H2: a ∈ l
P a ↔ Q a
  split; intros.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
l: list A
H1: filter P l = filter Q l
a: A
H2: a ∈ l
H3: P a
Q a
A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
l: list A
H1: filter P l = filter Q l
a: A
H2: a ∈ l
H3: Q a
P a
  -A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
l: list A
H1: filter P l = filter Q l
a: A
H2: a ∈ l
H3: P a
Q a by eapply elem_of_list_filter; rewrite <- H1; apply elem_of_list_filter.
  -A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
l: list A
H1: filter P l = filter Q l
a: A
H2: a ∈ l
H3: Q a
P a by eapply elem_of_list_filter; rewrite H1; apply elem_of_list_filter.
Qed.

Lemma filter_complement {X} P Q
 `{forall (x : X), Decision (P x)} `{forall (x : X), Decision (Q x)}
 (l : list X) :
 filter P l = filter Q l <->
 filter (fun x => ~ P x) l = filter (fun x => ~ Q x) l.X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
filter P l = filter Q l
↔ filter (λ x : X, ¬ P x) l =
  filter (λ x : X, ¬ Q x) l
Proof.X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
filter P l = filter Q l
↔ filter (λ x : X, ¬ P x) l =
  filter (λ x : X, ¬ Q x) l
  split; intros.X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter P l = filter Q l
filter (λ x : X, ¬ P x) l = filter (λ x : X, ¬ Q x) l
X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter (λ x : X, ¬ P x) l =
filter (λ x : X, ¬ Q x) l
filter P l = filter Q l
  -X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter P l = filter Q l
filter (λ x : X, ¬ P x) l = filter (λ x : X, ¬ Q x) l specialize (ext_elem_of_filter P Q l H1) as Hext.X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter P l = filter Q l
Hext: ∀ a : X, a ∈ l → P a ↔ Q a
filter (λ x : X, ¬ P x) l = filter (λ x : X, ¬ Q x) l
    apply filter_ext_elem_of.X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter P l = filter Q l
Hext: ∀ a : X, a ∈ l → P a ↔ Q a
∀ a : X, a ∈ l → ¬ P a ↔ ¬ Q a
    intros.X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter P l = filter Q l
Hext: ∀ a : X, a ∈ l → P a ↔ Q a
a: X
H2: a ∈ l
¬ P a ↔ ¬ Q a
    specialize (Hext a H2).X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter P l = filter Q l
a: X
Hext: P a ↔ Q a
H2: a ∈ l
¬ P a ↔ ¬ Q a
    by rewrite Hext; itauto.
  -X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter (λ x : X, ¬ P x) l =
filter (λ x : X, ¬ Q x) l
filter P l = filter Q l apply filter_ext_elem_of.X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter (λ x : X, ¬ P x) l =
filter (λ x : X, ¬ Q x) l
∀ a : X, a ∈ l → P a ↔ Q a intros.X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter (λ x : X, ¬ P x) l =
filter (λ x : X, ¬ Q x) l
a: X
H2: a ∈ l
P a ↔ Q a
    specialize (ext_elem_of_filter _ _ l H1 a H2) as Hext; cbn in Hext.X: Type
P, Q: X → Prop
H: ∀ x : X, Decision (P x)
H0: ∀ x : X, Decision (Q x)
l: list X
H1: filter (λ x : X, ¬ P x) l =
filter (λ x : X, ¬ Q x) l
a: X
H2: a ∈ l
Hext: ¬ P a ↔ ¬ Q a
P a ↔ Q a
    by destruct (decide (P a)), (decide (Q a)); itauto.
Qed.

Lemma NoDup_elem_of_remove A (l l' : list A) a :
  NoDup (l ++ a :: l') -> NoDup (l ++ l') /\ a ∉ l ++ l'.A: Type
l, l': list A
a: A
NoDup (l ++ a :: l') → NoDup (l ++ l') ∧ a ∉ l ++ l'
Proof.A: Type
l, l': list A
a: A
NoDup (l ++ a :: l') → NoDup (l ++ l') ∧ a ∉ l ++ l'
  intros Hnda.A: Type
l, l': list A
a: A
Hnda: NoDup (l ++ a :: l')
NoDup (l ++ l') ∧ a ∉ l ++ l'
  apply NoDup_app in Hnda.A: Type
l, l': list A
a: A
Hnda: NoDup l
∧ (∀ x : A, x ∈ l → x ∉ a :: l')
  ∧ NoDup (a :: l')
NoDup (l ++ l') ∧ a ∉ l ++ l'
  destruct Hnda as [Hnd [Ha Hnda]].A: Type
l, l': list A
a: A
Hnd: NoDup l
Ha: ∀ x : A, x ∈ l → x ∉ a :: l'
Hnda: NoDup (a :: l')
NoDup (l ++ l') ∧ a ∉ l ++ l'
  apply NoDup_cons in Hnda.A: Type
l, l': list A
a: A
Hnd: NoDup l
Ha: ∀ x : A, x ∈ l → x ∉ a :: l'
Hnda: (a ∉ l') ∧ NoDup l'
NoDup (l ++ l') ∧ a ∉ l ++ l'
  setoid_rewrite elem_of_cons in Ha.A: Type
l, l': list A
a: A
Hnd: NoDup l
Ha: ∀ x : A, x ∈ l → ¬ (x = a ∨ x ∈ l')
Hnda: (a ∉ l') ∧ NoDup l'
NoDup (l ++ l') ∧ a ∉ l ++ l'
  destruct Hnda as [Ha' Hnd']; split.A: Type
l, l': list A
a: A
Hnd: NoDup l
Ha: ∀ x : A, x ∈ l → ¬ (x = a ∨ x ∈ l')
Ha': a ∉ l'
Hnd': NoDup l'
NoDup (l ++ l')
A: Type
l, l': list A
a: A
Hnd: NoDup l
Ha: ∀ x : A, x ∈ l → ¬ (x = a ∨ x ∈ l')
Ha': a ∉ l'
Hnd': NoDup l'
a ∉ l ++ l'
  -A: Type
l, l': list A
a: A
Hnd: NoDup l
Ha: ∀ x : A, x ∈ l → ¬ (x = a ∨ x ∈ l')
Ha': a ∉ l'
Hnd': NoDup l'
NoDup (l ++ l') by apply NoDup_app; firstorder.
  -A: Type
l, l': list A
a: A
Hnd: NoDup l
Ha: ∀ x : A, x ∈ l → ¬ (x = a ∨ x ∈ l')
Ha': a ∉ l'
Hnd': NoDup l'
a ∉ l ++ l' by rewrite elem_of_app; firstorder.
Qed.

Lemma list_lookup_lt [A] (is : list A) :
  forall i, is_Some (is !! i) ->
  forall j, j < i -> is_Some (is !! j).A: Type
is: list A
∀ i : nat,
  is_Some (is !! i)
  → ∀ j : nat, j < i → is_Some (is !! j)
Proof.A: Type
is: list A
∀ i : nat,
  is_Some (is !! i)
  → ∀ j : nat, j < i → is_Some (is !! j)
  intros; apply lookup_lt_is_Some.A: Type
is: list A
i: nat
H: is_Some (is !! i)
j: nat
H0: j < i
j < length is
  by etransitivity; [| apply lookup_lt_is_Some].
Qed.

Lemma list_difference_singleton_not_in `{EqDecision A} :
  forall (l : list A) (a : A), a ∉ l ->
    list_difference l [a] = l.A: Type
EqDecision0: EqDecision A
∀ (l : list A) (a : A),
  a ∉ l → list_difference l [a] = l
Proof.A: Type
EqDecision0: EqDecision A
∀ (l : list A) (a : A),
  a ∉ l → list_difference l [a] = l
  intros l a; induction l; [done |].A: Type
EqDecision0: EqDecision A
a0: A
l: list A
a: A
IHl: a ∉ l → list_difference l [a] = l
a ∉ a0 :: l → list_difference (a0 :: l) [a] = a0 :: l
  rewrite not_elem_of_cons; intros [Hna0 Hnal]; cbn.A: Type
EqDecision0: EqDecision A
a0: A
l: list A
a: A
IHl: a ∉ l → list_difference l [a] = l
Hna0: a ≠ a0
Hnal: a ∉ l
(if decide_rel elem_of a0 [a]
 then list_difference l [a]
 else a0 :: list_difference l [a]) = a0 :: l
  case_decide as Ha0; [by apply elem_of_list_singleton in Ha0 |].A: Type
EqDecision0: EqDecision A
a0: A
l: list A
a: A
IHl: a ∉ l → list_difference l [a] = l
Hna0: a ≠ a0
Hnal: a ∉ l
Ha0: a0 ∉ [a]
a0 :: list_difference l [a] = a0 :: l
  by rewrite IHl.
Qed.

Lemma list_difference_singleton_length_in `{EqDecision A} :
  forall (l : list A) (a : A), a ∈ l ->
    length (list_difference l [a]) < length l.A: Type
EqDecision0: EqDecision A
∀ (l : list A) (a : A),
  a ∈ l → length (list_difference l [a]) < length l
Proof.A: Type
EqDecision0: EqDecision A
∀ (l : list A) (a : A),
  a ∈ l → length (list_difference l [a]) < length l
  intros l a; induction l; cbn; [by inversion 1 |].A: Type
EqDecision0: EqDecision A
a0: A
l: list A
a: A
IHl: a ∈ l
→ length (list_difference l [a]) < length l
a ∈ a0 :: l
→ length
    (if decide_rel elem_of a0 [a]
     then list_difference l [a]
     else a0 :: list_difference l [a]) < S (length l)
  case_decide as Ha0; rewrite elem_of_list_singleton in Ha0.A: Type
EqDecision0: EqDecision A
a0: A
l: list A
a: A
IHl: a ∈ l
→ length (list_difference l [a]) < length l
Ha0: a0 = a
a ∈ a0 :: l
→ length (list_difference l [a]) < S (length l)
A: Type
EqDecision0: EqDecision A
a0: A
l: list A
a: A
IHl: a ∈ l
→ length (list_difference l [a]) < length l
Ha0: a0 ≠ a
a ∈ a0 :: l
→ length (a0 :: list_difference l [a]) < S (length l)
  -A: Type
EqDecision0: EqDecision A
a0: A
l: list A
a: A
IHl: a ∈ l
→ length (list_difference l [a]) < length l
Ha0: a0 = a
a ∈ a0 :: l
→ length (list_difference l [a]) < S (length l) subst; intros _.A: Type
EqDecision0: EqDecision A
l: list A
a: A
IHl: a ∈ l
→ length (list_difference l [a]) < length l
length (list_difference l [a]) < S (length l)
    destruct (decide (a ∈ l)).A: Type
EqDecision0: EqDecision A
l: list A
a: A
IHl: a ∈ l
→ length (list_difference l [a]) < length l
e: a ∈ l
length (list_difference l [a]) < S (length l)
A: Type
EqDecision0: EqDecision A
l: list A
a: A
IHl: a ∈ l
→ length (list_difference l [a]) < length l
n: a ∉ l
length (list_difference l [a]) < S (length l)
    +A: Type
EqDecision0: EqDecision A
l: list A
a: A
IHl: a ∈ l
→ length (list_difference l [a]) < length l
e: a ∈ l
length (list_difference l [a]) < S (length l) by etransitivity; [apply IHl | lia].
    +A: Type
EqDecision0: EqDecision A
l: list A
a: A
IHl: a ∈ l
→ length (list_difference l [a]) < length l
n: a ∉ l
length (list_difference l [a]) < S (length l) by rewrite list_difference_singleton_not_in; [lia |].
  -A: Type
EqDecision0: EqDecision A
a0: A
l: list A
a: A
IHl: a ∈ l
→ length (list_difference l [a]) < length l
Ha0: a0 ≠ a
a ∈ a0 :: l
→ length (a0 :: list_difference l [a]) < S (length l) by inversion 1; subst; [done |]; cbn; spec IHl; [| lia].
Qed.

Lemma longer_subseteq_has_dups `{EqDecision A} :
  forall l1 l2 : list A, l1 ⊆ l2 -> length l1 > length l2 ->
  exists (i1 i2 : nat) (a : A), i1 <> i2 /\ l1 !! i1 = Some a /\ l1 !! i2 = Some a.A: Type
EqDecision0: EqDecision A
∀ l1 l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
Proof.A: Type
EqDecision0: EqDecision A
∀ l1 l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
  induction l1; [by inversion 2 |].A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
∀ l2 : list A,
  a :: l1 ⊆ l2
  → length (a :: l1) > length l2
    → ∃ (i1 i2 : nat) (a0 : A),
        i1 ≠ i2
        ∧ (a :: l1) !! i1 = Some a0
          ∧ (a :: l1) !! i2 = Some a0
  intros l2 Hl12 Hlen12.A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
∃ (i1 i2 : nat) (a0 : A),
  i1 ≠ i2
  ∧ (a :: l1) !! i1 = Some a0
    ∧ (a :: l1) !! i2 = Some a0
  destruct (decide (a ∈ l1)).A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
e: a ∈ l1
∃ (i1 i2 : nat) (a0 : A),
  i1 ≠ i2
  ∧ (a :: l1) !! i1 = Some a0
    ∧ (a :: l1) !! i2 = Some a0
A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
n: a ∉ l1
∃ (i1 i2 : nat) (a0 : A),
  i1 ≠ i2
  ∧ (a :: l1) !! i1 = Some a0
    ∧ (a :: l1) !! i2 = Some a0
  -A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
e: a ∈ l1
∃ (i1 i2 : nat) (a0 : A),
  i1 ≠ i2
  ∧ (a :: l1) !! i1 = Some a0
    ∧ (a :: l1) !! i2 = Some a0 exists 0.A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
e: a ∈ l1
∃ (i2 : nat) (a0 : A),
  0 ≠ i2
  ∧ (a :: l1) !! 0 = Some a0
    ∧ (a :: l1) !! i2 = Some a0
    apply elem_of_list_lookup_1 in e as [i2 Hi2].A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
i2: nat
Hi2: l1 !! i2 = Some a
∃ (i2 : nat) (a0 : A),
  0 ≠ i2
  ∧ (a :: l1) !! 0 = Some a0
    ∧ (a :: l1) !! i2 = Some a0
    by exists (S i2), a.
  -A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
n: a ∉ l1
∃ (i1 i2 : nat) (a0 : A),
  i1 ≠ i2
  ∧ (a :: l1) !! i1 = Some a0
    ∧ (a :: l1) !! i2 = Some a0 edestruct (IHl1 (list_difference l2 [a]))
           as (i1 & i2 & a' & Hi12 & Hli1 & Hli2); cycle 2.A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
n: a ∉ l1
i1, i2: nat
a': A
Hi12: i1 ≠ i2
Hli1: l1 !! i1 = Some a'
Hli2: l1 !! i2 = Some a'
∃ (i1 i2 : nat) (a0 : A),
  i1 ≠ i2
  ∧ (a :: l1) !! i1 = Some a0
    ∧ (a :: l1) !! i2 = Some a0
A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
n: a ∉ l1
l1 ⊆ list_difference l2 [a]
A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
n: a ∉ l1
length l1 > length (list_difference l2 [a])
    +A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
n: a ∉ l1
i1, i2: nat
a': A
Hi12: i1 ≠ i2
Hli1: l1 !! i1 = Some a'
Hli2: l1 !! i2 = Some a'
∃ (i1 i2 : nat) (a0 : A),
  i1 ≠ i2
  ∧ (a :: l1) !! i1 = Some a0
    ∧ (a :: l1) !! i2 = Some a0 by exists (S i1), (S i2), a'; cbn; itauto.
    +A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
n: a ∉ l1
l1 ⊆ list_difference l2 [a] intros x Hx.A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
n: a ∉ l1
x: A
Hx: x ∈ l1
x ∈ list_difference l2 [a]
      rewrite elem_of_list_difference, elem_of_list_singleton.A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
n: a ∉ l1
x: A
Hx: x ∈ l1
x ∈ l2 ∧ x ≠ a
      by split; [apply Hl12; right | by contradict n; subst].
    +A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: length (a :: l1) > length l2
n: a ∉ l1
length l1 > length (list_difference l2 [a]) cbn in Hlen12.A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: S (length l1) > length l2
n: a ∉ l1
length l1 > length (list_difference l2 [a])
      assert (Ha : a ∈ l2) by (apply Hl12; left).A: Type
EqDecision0: EqDecision A
a: A
l1: list A
IHl1: ∀ l2 : list A,
  l1 ⊆ l2
  → length l1 > length l2
    → ∃ (i1 i2 : nat) (a : A),
        i1 ≠ i2
        ∧ l1 !! i1 = Some a ∧ l1 !! i2 = Some a
l2: list A
Hl12: a :: l1 ⊆ l2
Hlen12: S (length l1) > length l2
n: a ∉ l1
Ha: a ∈ l2
length l1 > length (list_difference l2 [a])
      by specialize (list_difference_singleton_length_in _ _ Ha) as Hlen'; lia.
Qed.

Lemma ForAllSuffix2_lookup [A : Type] (R : A -> A -> Prop) l
  : ForAllSuffix2 R l <-> forall n a b, l !! n = Some a -> l !! (S n) = Some b -> R a b.A: Type
R: A → A → Prop
l: list A
ForAllSuffix2 R l
↔ (∀ (n : nat) (a b : A),
     l !! n = Some a → l !! S n = Some b → R a b)
Proof.A: Type
R: A → A → Prop
l: list A
ForAllSuffix2 R l
↔ (∀ (n : nat) (a b : A),
     l !! n = Some a → l !! S n = Some b → R a b)
  split.A: Type
R: A → A → Prop
l: list A
ForAllSuffix2 R l
→ ∀ (n : nat) (a b : A),
    l !! n = Some a → l !! S n = Some b → R a b
A: Type
R: A → A → Prop
l: list A
(∀ (n : nat) (a b : A),
   l !! n = Some a → l !! S n = Some b → R a b)
→ ForAllSuffix2 R l
  -A: Type
R: A → A → Prop
l: list A
ForAllSuffix2 R l
→ ∀ (n : nat) (a b : A),
    l !! n = Some a → l !! S n = Some b → R a b induction 1; cbn; [by inversion 2 |].A: Type
R: A → A → Prop
a: A
l: list A
H: match l with
| [] => True
| b :: _ => R a b
end
H0: ForAllSuffix
  (λ l : list A,
     match l with
     | [] => True
     | [a] => True
     | a :: b :: _ => R a b
     end) l
IHForAllSuffix: ∀ (n : nat) (a b : A),
  l !! n = Some a
  → l !! S n = Some b → R a b
∀ (n : nat) (a0 b : A),
  (a :: l) !! n = Some a0 → l !! n = Some b → R a0 b
    destruct n as [| n']; cbn.A: Type
R: A → A → Prop
a: A
l: list A
H: match l with
| [] => True
| b :: _ => R a b
end
H0: ForAllSuffix
  (λ l : list A,
     match l with
     | [] => True
     | [a] => True
     | a :: b :: _ => R a b
     end) l
IHForAllSuffix: ∀ (n : nat) (a b : A),
  l !! n = Some a
  → l !! S n = Some b → R a b
∀ a0 b : A,
  Some a = Some a0 → l !! 0 = Some b → R a0 b
A: Type
R: A → A → Prop
a: A
l: list A
H: match l with
| [] => True
| b :: _ => R a b
end
H0: ForAllSuffix
  (λ l : list A,
     match l with
     | [] => True
     | [a] => True
     | a :: b :: _ => R a b
     end) l
IHForAllSuffix: ∀ (n : nat) (a b : A),
  l !! n = Some a
  → l !! S n = Some b → R a b
n': nat
∀ a b : A,
  l !! n' = Some a → l !! S n' = Some b → R a b
    +A: Type
R: A → A → Prop
a: A
l: list A
H: match l with
| [] => True
| b :: _ => R a b
end
H0: ForAllSuffix
  (λ l : list A,
     match l with
     | [] => True
     | [a] => True
     | a :: b :: _ => R a b
     end) l
IHForAllSuffix: ∀ (n : nat) (a b : A),
  l !! n = Some a
  → l !! S n = Some b → R a b
∀ a0 b : A,
  Some a = Some a0 → l !! 0 = Some b → R a0 b by destruct l; do 2 inversion 1; subst.
    +A: Type
R: A → A → Prop
a: A
l: list A
H: match l with
| [] => True
| b :: _ => R a b
end
H0: ForAllSuffix
  (λ l : list A,
     match l with
     | [] => True
     | [a] => True
     | a :: b :: _ => R a b
     end) l
IHForAllSuffix: ∀ (n : nat) (a b : A),
  l !! n = Some a
  → l !! S n = Some b → R a b
n': nat
∀ a b : A,
  l !! n' = Some a → l !! S n' = Some b → R a b by intros; eapply IHForAllSuffix.
  -A: Type
R: A → A → Prop
l: list A
(∀ (n : nat) (a b : A),
   l !! n = Some a → l !! S n = Some b → R a b)
→ ForAllSuffix2 R l induction l as [| a [| b l']]; cbn.A: Type
R: A → A → Prop
(∀ (n : nat) (a b : A),
   [] !! n = Some a → None = Some b → R a b)
→ ForAllSuffix2 R []
A: Type
R: A → A → Prop
a: A
IHl: (∀ (n : nat) (a b : A), [] !! n = Some a → [] !! S n = Some b → R a b)
→ ForAllSuffix2 R []
(∀ (n : nat) (a0 b : A),
   [a] !! n = Some a0 → [] !! n = Some b → R a0 b)
→ ForAllSuffix2 R [a]
A: Type
R: A → A → Prop
a, b: A
l': list A
IHl: (∀ (n : nat) (a b0 : A),
(b :: l') !! n = Some a → (b :: l') !! S n = Some b0 → R a b0)
→ ForAllSuffix2 R (b :: l')
(∀ (n : nat) (a0 b0 : A),
   (a :: b :: l') !! n = Some a0
   → (b :: l') !! n = Some b0 → R a0 b0)
→ ForAllSuffix2 R (a :: b :: l')
    +A: Type
R: A → A → Prop
(∀ (n : nat) (a b : A),
   [] !! n = Some a → None = Some b → R a b)
→ ForAllSuffix2 R [] by constructor.
    +A: Type
R: A → A → Prop
a: A
IHl: (∀ (n : nat) (a b : A), [] !! n = Some a → [] !! S n = Some b → R a b)
→ ForAllSuffix2 R []
(∀ (n : nat) (a0 b : A),
   [a] !! n = Some a0 → [] !! n = Some b → R a0 b)
→ ForAllSuffix2 R [a] by repeat constructor.
    +A: Type
R: A → A → Prop
a, b: A
l': list A
IHl: (∀ (n : nat) (a b0 : A),
(b :: l') !! n = Some a → (b :: l') !! S n = Some b0 → R a b0)
→ ForAllSuffix2 R (b :: l')
(∀ (n : nat) (a0 b0 : A),
   (a :: b :: l') !! n = Some a0
   → (b :: l') !! n = Some b0 → R a0 b0)
→ ForAllSuffix2 R (a :: b :: l') constructor.A: Type
R: A → A → Prop
a, b: A
l': list A
IHl: (∀ (n : nat) (a b0 : A),
(b :: l') !! n = Some a → (b :: l') !! S n = Some b0 → R a b0)
→ ForAllSuffix2 R (b :: l')
H: ∀ (n : nat) (a0 b0 : A),
  (a :: b :: l') !! n = Some a0
  → (b :: l') !! n = Some b0 → R a0 b0
R a b
A: Type
R: A → A → Prop
a, b: A
l': list A
IHl: (∀ (n : nat) (a b0 : A),
(b :: l') !! n = Some a → (b :: l') !! S n = Some b0 → R a b0)
→ ForAllSuffix2 R (b :: l')
H: ∀ (n : nat) (a0 b0 : A),
  (a :: b :: l') !! n = Some a0
  → (b :: l') !! n = Some b0 → R a0 b0
ForAllSuffix
  (λ l : list A,
     match l with
     | [] => True
     | [a] => True
     | a :: b :: _ => R a b
     end) (b :: l')
      *A: Type
R: A → A → Prop
a, b: A
l': list A
IHl: (∀ (n : nat) (a b0 : A),
(b :: l') !! n = Some a → (b :: l') !! S n = Some b0 → R a b0)
→ ForAllSuffix2 R (b :: l')
H: ∀ (n : nat) (a0 b0 : A),
  (a :: b :: l') !! n = Some a0
  → (b :: l') !! n = Some b0 → R a0 b0
R a b by apply (H 0).
      *A: Type
R: A → A → Prop
a, b: A
l': list A
IHl: (∀ (n : nat) (a b0 : A),
(b :: l') !! n = Some a → (b :: l') !! S n = Some b0 → R a b0)
→ ForAllSuffix2 R (b :: l')
H: ∀ (n : nat) (a0 b0 : A),
  (a :: b :: l') !! n = Some a0
  → (b :: l') !! n = Some b0 → R a0 b0
ForAllSuffix
  (λ l : list A,
     match l with
     | [] => True
     | [a] => True
     | a :: b :: _ => R a b
     end) (b :: l') by apply IHl; intro n; apply (H (S n)).
Qed.

Lemma stdpp_nat_le_sum (x y : nat) : x <= y <-> exists z, y = x + z.x, y: nat
x ≤ y ↔ (∃ z : nat, y = x + z)
Proof.x, y: nat
x ≤ y ↔ (∃ z : nat, y = x + z)
  split.x, y: nat
x ≤ y → ∃ z : nat, y = x + z
x, y: nat
(∃ z : nat, y = x + z) → x ≤ y
  -x, y: nat
x ≤ y → ∃ z : nat, y = x + z by exists (y - x); lia.
  -x, y: nat
(∃ z : nat, y = x + z) → x ≤ y by intros [z ->]; lia.
Qed.

Lemma ForAllSuffix2_transitive_lookup
  [A : Type] (R : A -> A -> Prop) {HT : Transitive R} (l : list A)
  : ForAllSuffix2 R l <-> forall m n a b, m < n -> l !! m = Some a -> l !! n = Some b -> R a b.A: Type
R: A → A → Prop
HT: Transitive R
l: list A
ForAllSuffix2 R l
↔ (∀ (m n : nat) (a b : A),
     m < n → l !! m = Some a → l !! n = Some b → R a b)
Proof.A: Type
R: A → A → Prop
HT: Transitive R
l: list A
ForAllSuffix2 R l
↔ (∀ (m n : nat) (a b : A),
     m < n → l !! m = Some a → l !! n = Some b → R a b)
  rewrite ForAllSuffix2_lookup.A: Type
R: A → A → Prop
HT: Transitive R
l: list A
(∀ (n : nat) (a b : A),
   l !! n = Some a → l !! S n = Some b → R a b)
↔ (∀ (m n : nat) (a b : A),
     m < n → l !! m = Some a → l !! n = Some b → R a b)
  split; intro Hall; [| by intros n a b; apply Hall; lia].A: Type
R: A → A → Prop
HT: Transitive R
l: list A
Hall: ∀ (n : nat) (a b : A),
  l !! n = Some a → l !! S n = Some b → R a b
∀ (m n : nat) (a b : A),
  m < n → l !! m = Some a → l !! n = Some b → R a b
  intros m n a b Hlt.A: Type
R: A → A → Prop
HT: Transitive R
l: list A
Hall: ∀ (n : nat) (a b : A),
  l !! n = Some a → l !! S n = Some b → R a b
m, n: nat
a, b: A
Hlt: m < n
l !! m = Some a → l !! n = Some b → R a b
  apply stdpp_nat_le_sum in Hlt as [k ->]; rewrite Nat.add_comm.A: Type
R: A → A → Prop
HT: Transitive R
l: list A
Hall: ∀ (n : nat) (a b : A),
  l !! n = Some a → l !! S n = Some b → R a b
m: nat
a, b: A
k: nat
l !! m = Some a → l !! (k + S m) = Some b → R a b
  revert a b; induction k; cbn; [by apply Hall |].A: Type
R: A → A → Prop
HT: Transitive R
l: list A
Hall: ∀ (n : nat) (a b : A),
  l !! n = Some a → l !! S n = Some b → R a b
m, k: nat
IHk: ∀ a b : A,
  l !! m = Some a
  → l !! (k + S m) = Some b → R a b
∀ a b : A,
  l !! m = Some a → l !! S (k + S m) = Some b → R a b
  intros a b Ha Hb.A: Type
R: A → A → Prop
HT: Transitive R
l: list A
Hall: ∀ (n : nat) (a b : A),
  l !! n = Some a → l !! S n = Some b → R a b
m, k: nat
IHk: ∀ a b : A,
  l !! m = Some a
  → l !! (k + S m) = Some b → R a b
a, b: A
Ha: l !! m = Some a
Hb: l !! S (k + S m) = Some b
R a b
  assert (Hlt : k + S m < length l) by (apply lookup_lt_Some in Hb; lia).A: Type
R: A → A → Prop
HT: Transitive R
l: list A
Hall: ∀ (n : nat) (a b : A),
  l !! n = Some a → l !! S n = Some b → R a b
m, k: nat
IHk: ∀ a b : A,
  l !! m = Some a
  → l !! (k + S m) = Some b → R a b
a, b: A
Ha: l !! m = Some a
Hb: l !! S (k + S m) = Some b
Hlt: k + S m < length l
R a b
  apply lookup_lt_is_Some in Hlt as [c Hc].A: Type
R: A → A → Prop
HT: Transitive R
l: list A
Hall: ∀ (n : nat) (a b : A),
  l !! n = Some a → l !! S n = Some b → R a b
m, k: nat
IHk: ∀ a b : A,
  l !! m = Some a
  → l !! (k + S m) = Some b → R a b
a, b: A
Ha: l !! m = Some a
Hb: l !! S (k + S m) = Some b
c: A
Hc: l !! (k + S m) = Some c
R a b
  by transitivity c; [apply IHk | eapply Hall].
Qed.

If the n-th element of l is x, then we can decompose long enough suffixes of l into x and a suffix shorter by 1.

Lemma lastn_length_cons :
  forall {A : Type} (n : nat) (l : list A) (x : A),
    l !! n = Some x -> lastn (length l - n) l = x :: lastn (length l - S n) l.∀ (A : Type) (n : nat) (l : list A) (x : A),
  l !! n = Some x
  → lastn (length l - n) l =
    x :: lastn (length l - S n) l
Proof.∀ (A : Type) (n : nat) (l : list A) (x : A),
  l !! n = Some x
  → lastn (length l - n) l =
    x :: lastn (length l - S n) l
  intros A n l x H.A: Type
n: nat
l: list A
x: A
H: l !! n = Some x
lastn (length l - n) l = x :: lastn (length l - S n) l
  unfold lastn.A: Type
n: nat
l: list A
x: A
H: l !! n = Some x
rev (take (length l - n) (rev l)) =
x :: rev (take (length l - S n) (rev l))
  rewrite <- rev_length, <- !skipn_rev, rev_involutive.A: Type
n: nat
l: list A
x: A
H: l !! n = Some x
drop n l = x :: drop (S n) l
  by apply drop_S.
Qed.

Lemma list_subseteq_filter {A} P Q
  `{forall (x : A), Decision (P x)} `{forall (x : A), Decision (Q x)} :
  (forall a, P a -> Q a) ->
  forall s : list A, filter P s ⊆ filter Q s.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
(∀ a : A, P a → Q a)
→ ∀ s : list A, filter P s ⊆ filter Q s
Proof.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
(∀ a : A, P a → Q a)
→ ∀ s : list A, filter P s ⊆ filter Q s
  induction s; cbn; intros x Hin; [done |].A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, P a → Q a
a: A
s: list A
IHs: filter P s ⊆ filter Q s
x: A
Hin: x ∈ (if decide (P a) then a :: filter P s else filter P s)
x
∈ (if decide (Q a)
   then a :: filter Q s
   else filter Q s)
  by destruct (decide (P a)), (decide (Q a)); cbn in *; rewrite ?elem_of_cons in *; itauto.
Qed.

Lemma filter_length_fn {A} P Q
  `{forall (x : A), Decision (P x)} `{forall (x : A), Decision (Q x)}
  s (Hfg : Forall (fun a => P a -> Q a) s) :
  length (filter P s) <= length (filter Q s).A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
s: list A
Hfg: Forall (λ a : A, P a → Q a) s
length (filter P s) ≤ length (filter Q s)
Proof.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
s: list A
Hfg: Forall (λ a : A, P a → Q a) s
length (filter P s) ≤ length (filter Q s)
  induction s; simpl; [lia |].A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
a: A
s: list A
Hfg: Forall (λ a : A, P a → Q a) (a :: s)
IHs: Forall (λ a : A, P a → Q a) s → length (filter P s) ≤ length (filter Q s)
length (filter P (a :: s))
≤ length (filter Q (a :: s))
  inversion Hfg; subst.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
a: A
s: list A
Hfg: Forall (λ a : A, P a → Q a) (a :: s)
IHs: Forall (λ a : A, P a → Q a) s → length (filter P s) ≤ length (filter Q s)
H3: P a → Q a
H4: Forall (λ a : A, P a → Q a) s
length (filter P (a :: s))
≤ length (filter Q (a :: s)) specialize (IHs H4).A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
a: A
s: list A
Hfg: Forall (λ a : A, P a → Q a) (a :: s)
IHs: length (filter P s) ≤ length (filter Q s)
H3: P a → Q a
H4: Forall (λ a : A, P a → Q a) s
length (filter P (a :: s))
≤ length (filter Q (a :: s))
  rewrite 2 filter_cons.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
a: A
s: list A
Hfg: Forall (λ a : A, P a → Q a) (a :: s)
IHs: length (filter P s) ≤ length (filter Q s)
H3: P a → Q a
H4: Forall (λ a : A, P a → Q a) s
length
  (if decide (P a)
   then a :: filter P s
   else filter P s)
≤ length
    (if decide (Q a)
     then a :: filter Q s
     else filter Q s)
  by destruct (decide (P a)), (decide (Q a)); cbn; itauto lia.
Qed.

Lemma nth_error_filter
  {A} P `{forall (x : A), Decision (P x)}
  (l : list A)
  (n : nat)
  (a : A)
  (Hnth : nth_error (filter P l) n = Some a)
  : exists (nth : nat),
    nth_error_filter_index P l n = Some nth
    /\ nth_error l nth = Some a.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
n: nat
a: A
Hnth: nth_error (filter P l) n = Some a
∃ nth : nat,
  nth_error_filter_index P l n = Some nth
  ∧ nth_error l nth = Some a
Proof.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
n: nat
a: A
Hnth: nth_error (filter P l) n = Some a
∃ nth : nat,
  nth_error_filter_index P l n = Some nth
  ∧ nth_error l nth = Some a
  generalize dependent a.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
n: nat
∀ a : A,
  nth_error (filter P l) n = Some a
  → ∃ nth : nat,
      nth_error_filter_index P l n = Some nth
      ∧ nth_error l nth = Some a generalize dependent n.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
∀ (n : nat) (a : A),
  nth_error (filter P l) n = Some a
  → ∃ nth : nat,
      nth_error_filter_index P l n = Some nth
      ∧ nth_error l nth = Some a
  induction l.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
∀ (n : nat) (a : A),
  nth_error (filter P []) n = Some a
  → ∃ nth : nat,
      nth_error_filter_index P [] n = Some nth
      ∧ nth_error [] nth = Some a
A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
∀ (n : nat) (a0 : A),
  nth_error (filter P (a :: l)) n = Some a0
  → ∃ nth : nat,
      nth_error_filter_index P (a :: l) n = Some nth
      ∧ nth_error (a :: l) nth = Some a0
  -A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
∀ (n : nat) (a : A),
  nth_error (filter P []) n = Some a
  → ∃ nth : nat,
      nth_error_filter_index P [] n = Some nth
      ∧ nth_error [] nth = Some a by intros []; cbn; inversion 1.
  -A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
∀ (n : nat) (a0 : A),
  nth_error (filter P (a :: l)) n = Some a0
  → ∃ nth : nat,
      nth_error_filter_index P (a :: l) n = Some nth
      ∧ nth_error (a :: l) nth = Some a0 intros.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
n: nat
a0: A
Hnth: nth_error (filter P (a :: l)) n = Some a0
∃ nth : nat,
  nth_error_filter_index P (a :: l) n = Some nth
  ∧ nth_error (a :: l) nth = Some a0 rewrite filter_cons in Hnth.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
n: nat
a0: A
Hnth: nth_error
  (if decide (P a)
   then a :: filter P l
   else filter P l) n = Some a0
∃ nth : nat,
  nth_error_filter_index P (a :: l) n = Some nth
  ∧ nth_error (a :: l) nth = Some a0 simpl.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
n: nat
a0: A
Hnth: nth_error
  (if decide (P a)
   then a :: filter P l
   else filter P l) n = Some a0
∃ nth : nat,
  (if decide (P a)
   then
    match n with
    | 0 => Some 0
    | S n' =>
        option_map S (nth_error_filter_index P l n')
    end
   else option_map S (nth_error_filter_index P l n)) =
  Some nth ∧ nth_error (a :: l) nth = Some a0 destruct (decide (P a)).A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
n: nat
a0: A
p: P a
Hnth: nth_error (a :: filter P l) n = Some a0
∃ nth : nat,
  match n with
  | 0 => Some 0
  | S n' =>
      option_map S (nth_error_filter_index P l n')
  end = Some nth ∧ nth_error (a :: l) nth = Some a0
A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
  nth_error (filter P l) n = Some a
  → ∃ nth : nat,
      nth_error_filter_index P l n = Some nth
      ∧ nth_error l nth = Some a
n: nat
a0: A
n0: ¬ P a
Hnth: nth_error (filter P l) n = Some a0
∃ nth : nat,
  option_map S (nth_error_filter_index P l n) =
  Some nth ∧ nth_error (a :: l) nth = Some a0
    +A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
n: nat
a0: A
p: P a
Hnth: nth_error (a :: filter P l) n = Some a0
∃ nth : nat,
  match n with
  | 0 => Some 0
  | S n' =>
      option_map S (nth_error_filter_index P l n')
  end = Some nth ∧ nth_error (a :: l) nth = Some a0 destruct n.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
a0: A
p: P a
Hnth: nth_error (a :: filter P l) 0 = Some a0
∃ nth : nat,
  Some 0 = Some nth ∧ nth_error (a :: l) nth = Some a0
A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
  nth_error (filter P l) n = Some a
  → ∃ nth : nat,
      nth_error_filter_index P l n = Some nth
      ∧ nth_error l nth = Some a
n: nat
a0: A
p: P a
Hnth: nth_error (a :: filter P l) (S n) = Some a0
∃ nth : nat,
  option_map S (nth_error_filter_index P l n) =
  Some nth ∧ nth_error (a :: l) nth = Some a0
      *A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
a0: A
p: P a
Hnth: nth_error (a :: filter P l) 0 = Some a0
∃ nth : nat,
  Some 0 = Some nth ∧ nth_error (a :: l) nth = Some a0 by inversion Hnth; subst; exists 0.
      *A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
n: nat
a0: A
p: P a
Hnth: nth_error (a :: filter P l) (S n) = Some a0
∃ nth : nat,
  option_map S (nth_error_filter_index P l n) =
  Some nth ∧ nth_error (a :: l) nth = Some a0 simpl in Hnth.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
n: nat
a0: A
p: P a
Hnth: nth_error (filter P l) n = Some a0
∃ nth : nat,
  option_map S (nth_error_filter_index P l n) =
  Some nth ∧ nth_error (a :: l) nth = Some a0
        specialize (IHl n a0 Hnth).A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
n: nat
a0: A
IHl: ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a0
p: P a
Hnth: nth_error (filter P l) n = Some a0
∃ nth : nat,
  option_map S (nth_error_filter_index P l n) =
  Some nth ∧ nth_error (a :: l) nth = Some a0
        destruct IHl as [nth [Hnth' Ha0]].A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
n: nat
a0: A
nth: nat
Hnth': nth_error_filter_index P l n = Some nth
Ha0: nth_error l nth = Some a0
p: P a
Hnth: nth_error (filter P l) n = Some a0
∃ nth : nat,
  option_map S (nth_error_filter_index P l n) =
  Some nth ∧ nth_error (a :: l) nth = Some a0
        exists (S nth).A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
n: nat
a0: A
nth: nat
Hnth': nth_error_filter_index P l n = Some nth
Ha0: nth_error l nth = Some a0
p: P a
Hnth: nth_error (filter P l) n = Some a0
option_map S (nth_error_filter_index P l n) =
Some (S nth) ∧ nth_error (a :: l) (S nth) = Some a0
        by rewrite Hnth'.
    +A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
IHl: ∀ (n : nat) (a : A),
nth_error (filter P l) n = Some a
→ ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a
n: nat
a0: A
n0: ¬ P a
Hnth: nth_error (filter P l) n = Some a0
∃ nth : nat,
  option_map S (nth_error_filter_index P l n) =
  Some nth ∧ nth_error (a :: l) nth = Some a0 specialize (IHl n a0 Hnth).A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
n: nat
a0: A
IHl: ∃ nth : nat,
nth_error_filter_index P l n = Some nth ∧ nth_error l nth = Some a0
n0: ¬ P a
Hnth: nth_error (filter P l) n = Some a0
∃ nth : nat,
  option_map S (nth_error_filter_index P l n) =
  Some nth ∧ nth_error (a :: l) nth = Some a0
      destruct IHl as [nth [Hnth' Ha0]].A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
n: nat
a0: A
nth: nat
Hnth': nth_error_filter_index P l n = Some nth
Ha0: nth_error l nth = Some a0
n0: ¬ P a
Hnth: nth_error (filter P l) n = Some a0
∃ nth : nat,
  option_map S (nth_error_filter_index P l n) =
  Some nth ∧ nth_error (a :: l) nth = Some a0
      exists (S nth).A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
n: nat
a0: A
nth: nat
Hnth': nth_error_filter_index P l n = Some nth
Ha0: nth_error l nth = Some a0
n0: ¬ P a
Hnth: nth_error (filter P l) n = Some a0
option_map S (nth_error_filter_index P l n) =
Some (S nth) ∧ nth_error (a :: l) (S nth) = Some a0
      by rewrite Hnth'.
Qed.

Lemma filter_subseteq {A} P `{forall (x : A), Decision (P x)} (s1 s2 : list A) :
  s1 ⊆ s2 ->
  filter P s1 ⊆ filter P s2.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
s1, s2: list A
s1 ⊆ s2 → filter P s1 ⊆ filter P s2
Proof.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
s1, s2: list A
s1 ⊆ s2 → filter P s1 ⊆ filter P s2
  induction s1; intros; intro x; intros.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
s2: list A
H0: [] ⊆ s2
x: A
H1: x ∈ filter P []
x ∈ filter P s2
A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
H1: x ∈ filter P (a :: s1)
x ∈ filter P s2
  -A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
s2: list A
H0: [] ⊆ s2
x: A
H1: x ∈ filter P []
x ∈ filter P s2 by apply not_elem_of_nil in H1.
  -A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
H1: x ∈ filter P (a :: s1)
x ∈ filter P s2 rewrite filter_cons in H1.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
H1: x
∈ (if decide (P a)
   then a :: filter P s1
   else filter P s1)
x ∈ filter P s2
    destruct (decide (P a)).A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
p: P a
H1: x ∈ a :: filter P s1
x ∈ filter P s2
A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
n: ¬ P a
H1: x ∈ filter P s1
x ∈ filter P s2
    +A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
p: P a
H1: x ∈ a :: filter P s1
x ∈ filter P s2 rewrite elem_of_cons in H1.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
p: P a
H1: x = a ∨ x ∈ filter P s1
x ∈ filter P s2
      destruct H1.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
p: P a
H1: x = a
x ∈ filter P s2
A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
p: P a
H1: x ∈ filter P s1
x ∈ filter P s2
      *A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
p: P a
H1: x = a
x ∈ filter P s2 subst; apply elem_of_list_filter.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
p: P a
P a ∧ a ∈ s2
        by split; [| apply H0; left].
      *A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
p: P a
H1: x ∈ filter P s1
x ∈ filter P s2 apply IHs1; [| done].A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
p: P a
H1: x ∈ filter P s1
s1 ⊆ s2
        by intros y Hel; apply H0; right.
    +A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
n: ¬ P a
H1: x ∈ filter P s1
x ∈ filter P s2 apply IHs1; [| done].A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
s1, s2: list A
IHs1: s1 ⊆ s2 → filter P s1 ⊆ filter P s2
H0: a :: s1 ⊆ s2
x: A
n: ¬ P a
H1: x ∈ filter P s1
s1 ⊆ s2
      by intros y Hel; apply H0; right.
Qed.

Lemma filter_subseteq_fn {A} P Q
  `{forall (x : A), Decision (P x)} `{forall (x : A), Decision (Q x)} :
  (forall a, P a -> Q a) ->
  forall (s : list A), filter P s ⊆ filter Q s.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
(∀ a : A, P a → Q a)
→ ∀ s : list A, filter P s ⊆ filter Q s
Proof.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
(∀ a : A, P a → Q a)
→ ∀ s : list A, filter P s ⊆ filter Q s
  induction s; cbn; intros x H2; [by inversion H2 |].A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, P a → Q a
a: A
s: list A
IHs: filter P s ⊆ filter Q s
x: A
H2: x
∈ (if decide (P a)
   then a :: filter P s
   else filter P s)
x
∈ (if decide (Q a)
   then a :: filter Q s
   else filter Q s)
  destruct (decide (P a)), (decide (Q a)); rewrite ?elem_of_cons in H2.A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, P a → Q a
a: A
s: list A
IHs: filter P s ⊆ filter Q s
x: A
p: P a
H2: x = a ∨ x ∈ filter P s
q: Q a
x ∈ a :: filter Q s
A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, P a → Q a
a: A
s: list A
IHs: filter P s ⊆ filter Q s
x: A
p: P a
H2: x = a ∨ x ∈ filter P s
n: ¬ Q a
x ∈ filter Q s
A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, P a → Q a
a: A
s: list A
IHs: filter P s ⊆ filter Q s
x: A
n: ¬ P a
H2: x ∈ filter P s
q: Q a
x ∈ a :: filter Q s
A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, P a → Q a
a: A
s: list A
IHs: filter P s ⊆ filter Q s
x: A
n: ¬ P a
H2: x ∈ filter P s
n0: ¬ Q a
x ∈ filter Q s
  -A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, P a → Q a
a: A
s: list A
IHs: filter P s ⊆ filter Q s
x: A
p: P a
H2: x = a ∨ x ∈ filter P s
q: Q a
x ∈ a :: filter Q s by destruct H2 as [-> |]; [left | right; apply IHs].
  -A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, P a → Q a
a: A
s: list A
IHs: filter P s ⊆ filter Q s
x: A
p: P a
H2: x = a ∨ x ∈ filter P s
n: ¬ Q a
x ∈ filter Q s by itauto.
  -A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, P a → Q a
a: A
s: list A
IHs: filter P s ⊆ filter Q s
x: A
n: ¬ P a
H2: x ∈ filter P s
q: Q a
x ∈ a :: filter Q s by right; apply IHs.
  -A: Type
P, Q: A → Prop
H: ∀ x : A, Decision (P x)
H0: ∀ x : A, Decision (Q x)
H1: ∀ a : A, P a → Q a
a: A
s: list A
IHs: filter P s ⊆ filter Q s
x: A
n: ¬ P a
H2: x ∈ filter P s
n0: ¬ Q a
x ∈ filter Q s by itauto.
Qed.

Lemma Forall_filter_nil {A} P `{forall (x : A), Decision (P x)} l :
  Forall (fun a : A => ~ P a) l <-> filter P l = [].A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
Forall (λ a : A, ¬ P a) l ↔ filter P l = []
Proof.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
Forall (λ a : A, ¬ P a) l ↔ filter P l = []
  rewrite Forall_forall.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
(∀ x : A, x ∈ l → ¬ P x) ↔ filter P l = []
  split; intro Hnone.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
Hnone: ∀ x : A, x ∈ l → ¬ P x
filter P l = []
A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
Hnone: filter P l = []
∀ x : A, x ∈ l → ¬ P x
  -A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
Hnone: ∀ x : A, x ∈ l → ¬ P x
filter P l = [] induction l; cbn; [done |].A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
Hnone: ∀ x : A, x ∈ a :: l → ¬ P x
IHl: (∀ x : A, x ∈ l → ¬ P x) → filter P l = []
(if decide (P a) then a :: filter P l else filter P l) =
[]
    setoid_rewrite elem_of_cons in Hnone.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
a: A
l: list A
Hnone: ∀ x : A, x = a ∨ x ∈ l → ¬ P x
IHl: (∀ x : A, x ∈ l → ¬ P x) → filter P l = []
(if decide (P a) then a :: filter P l else filter P l) =
[]
    by rewrite decide_False; auto.
  -A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
Hnone: filter P l = []
∀ x : A, x ∈ l → ¬ P x intros x Hel Px.A: Type
P: A → Prop
H: ∀ x : A, Decision (P x)
l: list A
Hnone: filter P l = []
x: A
Hel: x ∈ l
Px: P x
False
    by eapply filter_nil_not_elem_of in Px.
Qed.

Lemma occurrences_ordering
  {A : Type}
  (a b : A)
  (la1 la2 lb1 lb2 : list A)
  (Heq : la1 ++ a :: la2 = lb1 ++ b :: lb2)
  (Ha : a ∉ b :: lb2)
  : exists lab : list A, lb1 = la1 ++ a :: lab.A: Type
a, b: A
la1, la2, lb1, lb2: list A
Heq: la1 ++ a :: la2 = lb1 ++ b :: lb2
Ha: a ∉ b :: lb2
∃ lab : list A, lb1 = la1 ++ a :: lab
Proof.A: Type
a, b: A
la1, la2, lb1, lb2: list A
Heq: la1 ++ a :: la2 = lb1 ++ b :: lb2
Ha: a ∉ b :: lb2
∃ lab : list A, lb1 = la1 ++ a :: lab
  generalize dependent lb2.A: Type
a, b: A
la1, la2, lb1: list A
∀ lb2 : list A,
  la1 ++ a :: la2 = lb1 ++ b :: lb2
  → a ∉ b :: lb2
    → ∃ lab : list A, lb1 = la1 ++ a :: lab generalize dependent la2.A: Type
a, b: A
la1, lb1: list A
∀ la2 lb2 : list A,
  la1 ++ a :: la2 = lb1 ++ b :: lb2
  → a ∉ b :: lb2
    → ∃ lab : list A, lb1 = la1 ++ a :: lab
  generalize dependent b.A: Type
a: A
la1, lb1: list A
∀ (b : A) (la2 lb2 : list A),
  la1 ++ a :: la2 = lb1 ++ b :: lb2
  → a ∉ b :: lb2
    → ∃ lab : list A, lb1 = la1 ++ a :: lab generalize dependent lb1.A: Type
a: A
la1: list A
∀ (lb1 : list A) (b : A) (la2 lb2 : list A),
  la1 ++ a :: la2 = lb1 ++ b :: lb2
  → a ∉ b :: lb2
    → ∃ lab : list A, lb1 = la1 ++ a :: lab
  generalize dependent a.A: Type
la1: list A
∀ (a : A) (lb1 : list A) (b : A) (la2 lb2 : list A),
  la1 ++ a :: la2 = lb1 ++ b :: lb2
  → a ∉ b :: lb2
    → ∃ lab : list A, lb1 = la1 ++ a :: lab
  induction la1; intros; destruct lb1 as [| b0 lb1]; simpl in *
  ; inversion Heq; subst.A: Type
b: A
lb2: list A
Ha: b ∉ b :: lb2
Heq: b :: lb2 = b :: lb2
∃ lab : list A, [] = b :: lab
A: Type
b0: A
lb1: list A
b: A
lb2: list A
Ha: b0 ∉ b :: lb2
Heq: b0 :: lb1 ++ b :: lb2 = b0 :: lb1 ++ b :: lb2
∃ lab : list A, b0 :: lb1 = b0 :: lab
A: Type
la1: list A
IHla1: ∀ (a : A) (lb1 : list A) 
  (b : A) (la2 lb2 : list A),
  la1 ++ a :: la2 = lb1 ++ b :: lb2
  → a ∉ b :: lb2
    → ∃ lab : list A, lb1 = la1 ++ a :: lab
a0, b: A
la2: list A
Ha: a0 ∉ b :: la1 ++ a0 :: la2
Heq: b :: la1 ++ a0 :: la2 = b :: la1 ++ a0 :: la2
∃ lab : list A, [] = b :: la1 ++ a0 :: lab
A: Type
la1: list A
IHla1: ∀ (a : A) (lb1 : list A) 
  (b : A) (la2 lb2 : list A),
  la1 ++ a :: la2 = lb1 ++ b :: lb2
  → a ∉ b :: lb2
    → ∃ lab : list A, lb1 = la1 ++ a :: lab
a0, b0: A
lb1: list A
b: A
la2, lb2: list A
Heq: b0 :: la1 ++ a0 :: la2 = b0 :: lb1 ++ b :: lb2
Ha: a0 ∉ b :: lb2
H1: la1 ++ a0 :: la2 = lb1 ++ b :: lb2
∃ lab : list A, b0 :: lb1 = b0 :: la1 ++ a0 :: lab
  -A: Type
b: A
lb2: list A
Ha: b ∉ b :: lb2
Heq: b :: lb2 = b :: lb2
∃ lab : list A, [] = b :: lab by contradict Ha; left.
  -A: Type
b0: A
lb1: list A
b: A
lb2: list A
Ha: b0 ∉ b :: lb2
Heq: b0 :: lb1 ++ b :: lb2 = b0 :: lb1 ++ b :: lb2
∃ lab : list A, b0 :: lb1 = b0 :: lab by exists lb1.
  -A: Type
la1: list A
IHla1: ∀ (a : A) (lb1 : list A) 
  (b : A) (la2 lb2 : list A),
  la1 ++ a :: la2 = lb1 ++ b :: lb2
  → a ∉ b :: lb2
    → ∃ lab : list A, lb1 = la1 ++ a :: lab
a0, b: A
la2: list A
Ha: a0 ∉ b :: la1 ++ a0 :: la2
Heq: b :: la1 ++ a0 :: la2 = b :: la1 ++ a0 :: la2
∃ lab : list A, [] = b :: la1 ++ a0 :: lab by contradict Ha; rewrite elem_of_cons, elem_of_app, elem_of_cons; auto.
  -A: Type
la1: list A
IHla1: ∀ (a : A) (lb1 : list A) 
  (b : A) (la2 lb2 : list A),
  la1 ++ a :: la2 = lb1 ++ b :: lb2
  → a ∉ b :: lb2
    → ∃ lab : list A, lb1 = la1 ++ a :: lab
a0, b0: A
lb1: list A
b: A
la2, lb2: list A
Heq: b0 :: la1 ++ a0 :: la2 = b0 :: lb1 ++ b :: lb2
Ha: a0 ∉ b :: lb2
H1: la1 ++ a0 :: la2 = lb1 ++ b :: lb2
∃ lab : list A, b0 :: lb1 = b0 :: la1 ++ a0 :: lab specialize (IHla1 a0 lb1 b la2 lb2 H1 Ha).A: Type
la1: list A
a0: A
lb1: list A
IHla1: ∃ lab : list A, lb1 = la1 ++ a0 :: lab
b0, b: A
la2, lb2: list A
Heq: b0 :: la1 ++ a0 :: la2 = b0 :: lb1 ++ b :: lb2
Ha: a0 ∉ b :: lb2
H1: la1 ++ a0 :: la2 = lb1 ++ b :: lb2
∃ lab : list A, b0 :: lb1 = b0 :: la1 ++ a0 :: lab
    destruct IHla1 as [la0b Hla0b].A: Type
la1: list A
a0: A
lb1, la0b: list A
Hla0b: lb1 = la1 ++ a0 :: la0b
b0, b: A
la2, lb2: list A
Heq: b0 :: la1 ++ a0 :: la2 = b0 :: lb1 ++ b :: lb2
Ha: a0 ∉ b :: lb2
H1: la1 ++ a0 :: la2 = lb1 ++ b :: lb2
∃ lab : list A, b0 :: lb1 = b0 :: la1 ++ a0 :: lab
    by exists la0b; subst.
Qed.

Lemma list_max_elem_of_exists
   (l : list nat)
   (nz : list_max l > 0) :
   list_max l ∈ l.l: list nat
nz: list_max l > 0
list_max l ∈ l
Proof.l: list nat
nz: list_max l > 0
list_max l ∈ l
  induction l; simpl in *; [lia |].a: nat
l: list nat
nz: a `max` list_max l > 0
IHl: list_max l > 0 → list_max l ∈ l
a `max` list_max l ∈ a :: l
  rewrite elem_of_cons.a: nat
l: list nat
nz: a `max` list_max l > 0
IHl: list_max l > 0 → list_max l ∈ l
a `max` list_max l = a ∨ a `max` list_max l ∈ l
  by destruct (Nat.max_spec_le a (list_max l)) as [[H ->] | [H ->]]; itauto lia.
Qed.

Lemma omap_subseteq
  {A B : Type}
  (f : A -> option B)
  (l1 l2 : list A)
  (Hincl : l1 ⊆ l2)
  : omap f l1 ⊆ omap f l2.A, B: Type
f: A → option B
l1, l2: list A
Hincl: l1 ⊆ l2
omap f l1 ⊆ omap f l2
Proof.A, B: Type
f: A → option B
l1, l2: list A
Hincl: l1 ⊆ l2
omap f l1 ⊆ omap f l2
  by intros b; rewrite !elem_of_list_omap; firstorder.
Qed.

Lemma elem_of_cat_option
  {A : Type}
  (l : list (option A))
  (a : A)
  : a ∈ cat_option l <-> exists b : option A, b ∈ l /\ b = Some a.A: Type
l: list (option A)
a: A
a ∈ cat_option l
↔ (∃ b : option A, b ∈ l ∧ b = Some a)
Proof.A: Type
l: list (option A)
a: A
a ∈ cat_option l
↔ (∃ b : option A, b ∈ l ∧ b = Some a)
  by apply elem_of_list_omap.
Qed.

Lemma list_max_elem_of_exists2
   (l : list nat)
   (Hne : l <> []) :
   list_max l ∈ l.l: list nat
Hne: l ≠ []
list_max l ∈ l
Proof.l: list nat
Hne: l ≠ []
list_max l ∈ l
  destruct (list_max l) eqn: eq_max.l: list nat
Hne: l ≠ []
eq_max: list_max l = 0
0 ∈ l
l: list nat
Hne: l ≠ []
n: nat
eq_max: list_max l = S n
S n ∈ l
  -l: list nat
Hne: l ≠ []
eq_max: list_max l = 0
0 ∈ l destruct l; [by itauto congruence |].n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
0 ∈ n :: l
    specialize (list_max_le (n :: l) 0) as Hle.n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: list_max (n :: l) ≤ 0
↔ Forall (λ k : nat, k ≤ 0) (n :: l)
0 ∈ n :: l
    destruct Hle as [Hle _].n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: list_max (n :: l) ≤ 0
→ Forall (λ k : nat, k ≤ 0) (n :: l)
0 ∈ n :: l
    rewrite eq_max in Hle.n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: 0 ≤ 0 → Forall (λ k : nat, k ≤ 0) (n :: l)
0 ∈ n :: l spec Hle.n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: 0 ≤ 0 → Forall (λ k : nat, k ≤ 0) (n :: l)
0 ≤ 0
n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: Forall (λ k : nat, k ≤ 0) (n :: l)
0 ∈ n :: l apply Nat.le_refl.n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: Forall (λ k : nat, k ≤ 0) (n :: l)
0 ∈ n :: l
    rewrite Forall_forall in Hle.n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: ∀ x : nat, x ∈ n :: l → x ≤ 0
0 ∈ n :: l
    specialize (Hle n).n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: n ∈ n :: l → n ≤ 0
0 ∈ n :: l spec Hle.n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: n ∈ n :: l → n ≤ 0
n ∈ n :: l
n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: n ≤ 0
0 ∈ n :: l left.n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: n ≤ 0
0 ∈ n :: l
    assert (Hn0 : n = 0) by lia.n: nat
l: list nat
Hne: n :: l ≠ []
eq_max: list_max (n :: l) = 0
Hle: n ≤ 0
Hn0: n = 0
0 ∈ n :: l
    by rewrite Hn0; left.
  -l: list nat
Hne: l ≠ []
n: nat
eq_max: list_max l = S n
S n ∈ l specialize (list_max_elem_of_exists l) as Hmax.l: list nat
Hne: l ≠ []
n: nat
eq_max: list_max l = S n
Hmax: list_max l > 0 → list_max l ∈ l
S n ∈ l
    by rewrite <- eq_max; itauto lia.
Qed.

Lemma mode_not_empty
  `{EqDecision A}
  (l : list A)
  (Hne : l <> []) :
  mode l <> [].A: Type
EqDecision0: EqDecision A
l: list A
Hne: l ≠ []
mode l ≠ []
Proof.A: Type
EqDecision0: EqDecision A
l: list A
Hne: l ≠ []
mode l ≠ []
  destruct l; [done |].A: Type
EqDecision0: EqDecision A
a: A
l: list A
Hne: a :: l ≠ []
mode (a :: l) ≠ []
  remember (a :: l) as l'.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
mode l' ≠ []
  remember (List.map (count_occ decide_eq l') l') as occurrences.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
mode l' ≠ []

  assert (Hmaxp : list_max occurrences > 0).A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
list_max occurrences > 0
A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
mode l' ≠ [] {A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
list_max occurrences > 0
    rewrite Heqoccurrences, Heql'; cbn.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
(if decide (a = a)
 then S (count_occ decide_eq l a)
 else count_occ decide_eq l a)
`max` foldr Init.Nat.max 0
        (map
           (λ x : A,
              if decide (a = x)
              then S (count_occ decide_eq l x)
              else count_occ decide_eq l x) l) > 0
    by rewrite decide_True; [lia |].
  }A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
mode l' ≠ []

  assert (exists a, (count_occ decide_eq l' a) = list_max occurrences).A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: ∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
mode l' ≠ [] {A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
    assert (list_max occurrences ∈ occurrences).A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
list_max occurrences ∈ occurrences
A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: list_max occurrences ∈ occurrences
∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
    {A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
list_max occurrences ∈ occurrences
      apply list_max_exists.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
occurrences ≠ []
      by destruct occurrences; [cbn in Hmaxp; lia |].
    }A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: list_max occurrences ∈ occurrences
∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
    rewrite Heqoccurrences, elem_of_list_fmap in H.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: ∃ y : A,
  list_max (map (count_occ decide_eq l') l') =
  count_occ decide_eq l' y ∧ 
  y ∈ l'
∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
    destruct H as (x & Heq & Hin).A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
x: A
Heq: list_max (map (count_occ decide_eq l') l') =
count_occ decide_eq l' x
Hin: x ∈ l'
∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
    by rewrite Heqoccurrences; eauto.
  }A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: ∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
mode l' ≠ []

  assert (exists a, a ∈ mode l').A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: ∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
∃ a : A, a ∈ mode l'
A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: ∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
H0: ∃ a : A, a ∈ mode l'
mode l' ≠ [] {A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: ∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
∃ a : A, a ∈ mode l'
    destruct H.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
x: A
H: count_occ decide_eq l' x = list_max occurrences
∃ a : A, a ∈ mode l'
    exists x.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
x: A
H: count_occ decide_eq l' x = list_max occurrences
x ∈ mode l'
    unfold mode.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
x: A
H: count_occ decide_eq l' x = list_max occurrences
x
∈ filter
    (λ a : A,
       count_occ decide_eq l' a =
       list_max (map (count_occ decide_eq l') l')) l'
    apply elem_of_list_filter.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
x: A
H: count_occ decide_eq l' x = list_max occurrences
count_occ decide_eq l' x =
list_max (map (count_occ decide_eq l') l') ∧ x ∈ l'
    specialize (count_occ_In decide_eq l' x); rewrite <- elem_of_list_In.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
x: A
H: count_occ decide_eq l' x = list_max occurrences
x ∈ l' ↔ count_occ decide_eq l' x > 0
→ count_occ decide_eq l' x =
  list_max (map (count_occ decide_eq l') l') ∧ x ∈ l'
    by itauto congruence.
  }A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: ∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
H0: ∃ a : A, a ∈ mode l'
mode l' ≠ []
  intros contra; rewrite contra in H0.A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: ∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
H0: ∃ a : A, a ∈ []
contra: mode l' = []
False
  destruct H0 as [? H0].A: Type
EqDecision0: EqDecision A
a: A
l, l': list A
Heql': l' = a :: l
Hne: l' ≠ []
occurrences: list nat
Heqoccurrences: occurrences =
map (count_occ decide_eq l') l'
Hmaxp: list_max occurrences > 0
H: ∃ a : A,
  count_occ decide_eq l' a = list_max occurrences
x: A
H0: x ∈ []
contra: mode l' = []
False
  by apply elem_of_nil in H0.
Qed.

When a list contains two elements, either they are equal or we can split the list into three parts separated by the elements (and this can be done in two ways, depending on the order of the elements).

Lemma elem_of_list_split_2 :
  forall {A : Type} (l : list A) (x y : A),
    x ∈ l -> y ∈ l ->
      x = y \/ exists l1 l2 l3 : list A,
        l = l1 ++ x :: l2 ++ y :: l3 \/ l = l1 ++ y :: l2 ++ x :: l3.∀ (A : Type) (l : list A) (x y : A),
  x ∈ l
  → y ∈ l
    → x = y
      ∨ (∃ l1 l2 l3 : list A,
           l = l1 ++ x :: l2 ++ y :: l3
           ∨ l = l1 ++ y :: l2 ++ x :: l3)
Proof.∀ (A : Type) (l : list A) (x y : A),
  x ∈ l
  → y ∈ l
    → x = y
      ∨ (∃ l1 l2 l3 : list A,
           l = l1 ++ x :: l2 ++ y :: l3
           ∨ l = l1 ++ y :: l2 ++ x :: l3)
  intros A x y l Hx Hy.A: Type
x: list A
y, l: A
Hx: y ∈ x
Hy: l ∈ x
y = l
∨ (∃ l1 l2 l3 : list A,
     x = l1 ++ y :: l2 ++ l :: l3
     ∨ x = l1 ++ l :: l2 ++ y :: l3)
  apply elem_of_list_split in Hx as (l1 & l2 & ->).A: Type
y, l: A
l1, l2: list A
Hy: l ∈ l1 ++ y :: l2
y = l
∨ (∃ l3 l4 l5 : list A,
     l1 ++ y :: l2 = l3 ++ y :: l4 ++ l :: l5
     ∨ l1 ++ y :: l2 = l3 ++ l :: l4 ++ y :: l5)
  rewrite elem_of_app, elem_of_cons in Hy.A: Type
y, l: A
l1, l2: list A
Hy: l ∈ l1 ∨ l = y ∨ l ∈ l2
y = l
∨ (∃ l3 l4 l5 : list A,
     l1 ++ y :: l2 = l3 ++ y :: l4 ++ l :: l5
     ∨ l1 ++ y :: l2 = l3 ++ l :: l4 ++ y :: l5)
  destruct Hy as [Hy | [-> | Hy]]; [| by left |].A: Type
y, l: A
l1, l2: list A
Hy: l ∈ l1
y = l
∨ (∃ l3 l4 l5 : list A,
     l1 ++ y :: l2 = l3 ++ y :: l4 ++ l :: l5
     ∨ l1 ++ y :: l2 = l3 ++ l :: l4 ++ y :: l5)
A: Type
y, l: A
l1, l2: list A
Hy: l ∈ l2
y = l
∨ (∃ l3 l4 l5 : list A,
     l1 ++ y :: l2 = l3 ++ y :: l4 ++ l :: l5
     ∨ l1 ++ y :: l2 = l3 ++ l :: l4 ++ y :: l5)
  -A: Type
y, l: A
l1, l2: list A
Hy: l ∈ l1
y = l
∨ (∃ l3 l4 l5 : list A,
     l1 ++ y :: l2 = l3 ++ y :: l4 ++ l :: l5
     ∨ l1 ++ y :: l2 = l3 ++ l :: l4 ++ y :: l5) apply elem_of_list_split in Hy as (l11 & l12 & ->).A: Type
y, l: A
l2, l11, l12: list A
y = l
∨ (∃ l1 l3 l4 : list A,
     (l11 ++ l :: l12) ++ y :: l2 =
     l1 ++ y :: l3 ++ l :: l4
     ∨ (l11 ++ l :: l12) ++ y :: l2 =
       l1 ++ l :: l3 ++ y :: l4)
    right; exists l11, l12, l2; right; cbn.A: Type
y, l: A
l2, l11, l12: list A
(l11 ++ l :: l12) ++ y :: l2 =
l11 ++ l :: l12 ++ y :: l2
    by rewrite <- app_assoc.
  -A: Type
y, l: A
l1, l2: list A
Hy: l ∈ l2
y = l
∨ (∃ l3 l4 l5 : list A,
     l1 ++ y :: l2 = l3 ++ y :: l4 ++ l :: l5
     ∨ l1 ++ y :: l2 = l3 ++ l :: l4 ++ y :: l5) apply elem_of_list_split in Hy as (l21 & l22 & ->).A: Type
y, l: A
l1, l21, l22: list A
y = l
∨ (∃ l2 l3 l4 : list A,
     l1 ++ y :: l21 ++ l :: l22 =
     l2 ++ y :: l3 ++ l :: l4
     ∨ l1 ++ y :: l21 ++ l :: l22 =
       l2 ++ l :: l3 ++ y :: l4)
    by right; exists l1, l21, l22; left; cbn.
Qed.

Lemma mjoin_app {A} (l1 l2 : list (list A)) :
  mjoin (l1 ++ l2) = mjoin l1 ++ mjoin l2.A: Type
l1, l2: list (list A)
mjoin (l1 ++ l2) = mjoin l1 ++ mjoin l2
Proof.A: Type
l1, l2: list (list A)
mjoin (l1 ++ l2) = mjoin l1 ++ mjoin l2
  induction l1; cbn; [done |].A: Type
a: list A
l1, l2: list (list A)
IHl1: mjoin (l1 ++ l2) = mjoin l1 ++ mjoin l2
a ++ mjoin (l1 ++ l2) = (a ++ mjoin l1) ++ mjoin l2
  replace (mjoin (l1 ++ l2)) with (mjoin l1 ++ mjoin l2).A: Type
a: list A
l1, l2: list (list A)
IHl1: mjoin (l1 ++ l2) = mjoin l1 ++ mjoin l2
a ++ mjoin l1 ++ mjoin l2 =
(a ++ mjoin l1) ++ mjoin l2
  by rewrite app_assoc.
Qed.

Lemma mbind_app `(f : A -> list B) (l1 l2 : list A) :
  mbind f (l1 ++ l2) = mbind f l1 ++ mbind f l2.A, B: Type
f: A → list B
l1, l2: list A
(l1 ++ l2) ≫= f = (l1 ≫= f) ++ l2 ≫= f
Proof.A, B: Type
f: A → list B
l1, l2: list A
(l1 ++ l2) ≫= f = (l1 ≫= f) ++ l2 ≫= f by induction l1; [| cbn; rewrite IHl1, app_assoc]. Qed.

Lemma mbind_nils :
  forall {A B : Type} (f : A -> list B) (l : list A),
    Forall (fun x : A => f x = []) l ->
      mbind f l = [].∀ (A B : Type) (f : A → list B) (l : list A),
  Forall (λ x : A, f x = []) l → l ≫= f = []
Proof.∀ (A B : Type) (f : A → list B) (l : list A),
  Forall (λ x : A, f x = []) l → l ≫= f = []
  induction 1; cbn; [done |].A, B: Type
f: A → list B
x: A
l: list A
H: f x = []
H0: Forall (λ x : A, f x = []) l
IHForall: l ≫= f = []
f x ++ l ≫= f = []
  by rewrite H, IHForall; cbn.
Qed.

Lemma list_subseteq_inv_app :
  forall {A : Type} (l1 l2 l3 : list A),
    l1 ++ l2 ⊆ l3 -> l1 ⊆ l3 /\ l2 ⊆ l3.∀ (A : Type) (l1 l2 l3 : list A),
  l1 ++ l2 ⊆ l3 → l1 ⊆ l3 ∧ l2 ⊆ l3
Proof.∀ (A : Type) (l1 l2 l3 : list A),
  l1 ++ l2 ⊆ l3 → l1 ⊆ l3 ∧ l2 ⊆ l3
  unfold subseteq, list_subseteq.∀ (A : Type) (l1 l2 l3 : list A),
  (∀ x : A, x ∈ l1 ++ l2 → x ∈ l3)
  → (∀ x : A, x ∈ l1 → x ∈ l3)
    ∧ (∀ x : A, x ∈ l2 → x ∈ l3)
  intros A l1 l2 l3 Hsub.A: Type
l1, l2, l3: list A
Hsub: ∀ x : A, x ∈ l1 ++ l2 → x ∈ l3
(∀ x : A, x ∈ l1 → x ∈ l3)
∧ (∀ x : A, x ∈ l2 → x ∈ l3)
  split; intros x Hin.A: Type
l1, l2, l3: list A
Hsub: ∀ x : A, x ∈ l1 ++ l2 → x ∈ l3
x: A
Hin: x ∈ l1
x ∈ l3
A: Type
l1, l2, l3: list A
Hsub: ∀ x : A, x ∈ l1 ++ l2 → x ∈ l3
x: A
Hin: x ∈ l2
x ∈ l3
  -A: Type
l1, l2, l3: list A
Hsub: ∀ x : A, x ∈ l1 ++ l2 → x ∈ l3
x: A
Hin: x ∈ l1
x ∈ l3 by apply Hsub, elem_of_app; left.
  -A: Type
l1, l2, l3: list A
Hsub: ∀ x : A, x ∈ l1 ++ l2 → x ∈ l3
x: A
Hin: x ∈ l2
x ∈ l3 by apply Hsub, elem_of_app; right.
Qed.

#[export] Instance sum_list_with_proper `(f : index -> nat) :
  Proper ((≡ₚ) ==> (=)) (sum_list_with f).index: Type
f: index → nat
Proper (Permutation ==> eq) (sum_list_with f)
Proof.index: Type
f: index → nat
Proper (Permutation ==> eq) (sum_list_with f)
  induction 1; cbn; [done | ..].index: Type
f: index → nat
x: index
l, l': list index
H: l ≡ₚ l'
IHPermutation: sum_list_with f l = sum_list_with f l'
f x + sum_list_with f l = f x + sum_list_with f l'
index: Type
f: index → nat
x, y: index
l: list index
f y + (f x + sum_list_with f l) =
f x + (f y + sum_list_with f l)
index: Type
f: index → nat
l, l', l'': list index
H: l ≡ₚ l'
H0: l' ≡ₚ l''
IHPermutation1: sum_list_with f l =
sum_list_with f l'
IHPermutation2: sum_list_with f l' =
sum_list_with f l''
sum_list_with f l = sum_list_with f l''
  -index: Type
f: index → nat
x: index
l, l': list index
H: l ≡ₚ l'
IHPermutation: sum_list_with f l = sum_list_with f l'
f x + sum_list_with f l = f x + sum_list_with f l' by rewrite IHPermutation.
  -index: Type
f: index → nat
x, y: index
l: list index
f y + (f x + sum_list_with f l) =
f x + (f y + sum_list_with f l) by lia.
  -index: Type
f: index → nat
l, l', l'': list index
H: l ≡ₚ l'
H0: l' ≡ₚ l''
IHPermutation1: sum_list_with f l =
sum_list_with f l'
IHPermutation2: sum_list_with f l' =
sum_list_with f l''
sum_list_with f l = sum_list_with f l'' by congruence.
Qed.

Lemma sum_list_with_ext_forall index (f g : index -> nat) (l : list index) :
  (forall (i : index), i ∈ l -> f i = g i) ->
    sum_list_with f l = sum_list_with g l.index: Type
f, g: index → nat
l: list index
(∀ i : index, i ∈ l → f i = g i)
→ sum_list_with f l = sum_list_with g l
Proof.index: Type
f, g: index → nat
l: list index
(∀ i : index, i ∈ l → f i = g i)
→ sum_list_with f l = sum_list_with g l
  induction l; cbn; intros Heq; [done |].index: Type
f, g: index → nat
a: index
l: list index
IHl: (∀ i : index, i ∈ l → f i = g i)
→ sum_list_with f l = sum_list_with g l
Heq: ∀ i : index, i ∈ a :: l → f i = g i
f a + sum_list_with f l = g a + sum_list_with g l
  rewrite Heq by left.index: Type
f, g: index → nat
a: index
l: list index
IHl: (∀ i : index, i ∈ l → f i = g i)
→ sum_list_with f l = sum_list_with g l
Heq: ∀ i : index, i ∈ a :: l → f i = g i
g a + sum_list_with f l = g a + sum_list_with g l
  rewrite IHl; [done |].index: Type
f, g: index → nat
a: index
l: list index
IHl: (∀ i : index, i ∈ l → f i = g i)
→ sum_list_with f l = sum_list_with g l
Heq: ∀ i : index, i ∈ a :: l → f i = g i
∀ i : index, i ∈ l → f i = g i
  by intros; apply Heq; right.
Qed.

Lemma sum_list_with_zero `(f : index -> nat) (l : list index) :
  sum_list_with  f l = 0 <-> forall (i : index), i ∈ l -> f i = 0.index: Type
f: index → nat
l: list index
sum_list_with f l = 0 ↔ (∀ i : index, i ∈ l → f i = 0)
Proof.index: Type
f: index → nat
l: list index
sum_list_with f l = 0 ↔ (∀ i : index, i ∈ l → f i = 0)
  split.index: Type
f: index → nat
l: list index
sum_list_with f l = 0 → ∀ i : index, i ∈ l → f i = 0
index: Type
f: index → nat
l: list index
(∀ i : index, i ∈ l → f i = 0) → sum_list_with f l = 0
  -index: Type
f: index → nat
l: list index
sum_list_with f l = 0 → ∀ i : index, i ∈ l → f i = 0 intros Hsum i Hi.index: Type
f: index → nat
l: list index
Hsum: sum_list_with f l = 0
i: index
Hi: i ∈ l
f i = 0
    apply sum_list_with_in with (f := f) in Hi.index: Type
f: index → nat
l: list index
Hsum: sum_list_with f l = 0
i: index
Hi: f i ≤ sum_list_with f l
f i = 0
    by lia.
  -index: Type
f: index → nat
l: list index
(∀ i : index, i ∈ l → f i = 0) → sum_list_with f l = 0 induction l; intros Hall; cbn; [done |].index: Type
f: index → nat
a: index
l: list index
IHl: (∀ i : index, i ∈ l → f i = 0)
→ sum_list_with f l = 0
Hall: ∀ i : index, i ∈ a :: l → f i = 0
f a + sum_list_with f l = 0
    rewrite Hall by left.index: Type
f: index → nat
a: index
l: list index
IHl: (∀ i : index, i ∈ l → f i = 0)
→ sum_list_with f l = 0
Hall: ∀ i : index, i ∈ a :: l → f i = 0
0 + sum_list_with f l = 0
    rewrite IHl; [done |].index: Type
f: index → nat
a: index
l: list index
IHl: (∀ i : index, i ∈ l → f i = 0)
→ sum_list_with f l = 0
Hall: ∀ i : index, i ∈ a :: l → f i = 0
∀ i : index, i ∈ l → f i = 0
    by intros; apply Hall; right.
Qed.

Lemma dsig_NoDup_map `(P : A -> Prop) `{Pdec : forall a, Decision (P a)} :
  forall (l : list (dsig P)),
    NoDup l <-> NoDup (map proj1_sig l).A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
∀ l : list (dsig P), NoDup l ↔ NoDup (map proj1_sig l)
Proof.A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
∀ l : list (dsig P), NoDup l ↔ NoDup (map proj1_sig l)
  split.A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
l: list (dsig P)
NoDup l → NoDup (map proj1_sig l)
A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
l: list (dsig P)
NoDup (map proj1_sig l) → NoDup l
  -A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
l: list (dsig P)
NoDup l → NoDup (map proj1_sig l) induction 1 as [| da dl Hda]; cbn; constructor; [| done].A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
da: dsig P
dl: list (dsig P)
Hda: da ∉ dl
H: NoDup dl
IHNoDup: NoDup (map proj1_sig dl)
`da ∉ map proj1_sig dl
    rewrite elem_of_list_fmap.A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
da: dsig P
dl: list (dsig P)
Hda: da ∉ dl
H: NoDup dl
IHNoDup: NoDup (map proj1_sig dl)
¬ (∃ y : {x : A | bool_decide (P x)},
     `da = `y ∧ y ∈ dl)
    intros (_da & Heq & H_da).A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
da: dsig P
dl: list (dsig P)
Hda: da ∉ dl
H: NoDup dl
IHNoDup: NoDup (map proj1_sig dl)
_da: {x : A | bool_decide (P x)}
Heq: `da = `_da
H_da: _da ∈ dl
False
    by apply dsig_eq in Heq as <-.
  -A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
l: list (dsig P)
NoDup (map proj1_sig l) → NoDup l induction l; cbn; [by constructor |].A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
a: dsig P
l: list (dsig P)
IHl: NoDup (map proj1_sig l) → NoDup l
NoDup (`a :: map proj1_sig l) → NoDup (a :: l)
    rewrite !NoDup_cons.A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
a: dsig P
l: list (dsig P)
IHl: NoDup (map proj1_sig l) → NoDup l
(`a ∉ map proj1_sig l) ∧ NoDup (map proj1_sig l)
→ (a ∉ l) ∧ NoDup l
    intros [Ha ?]; split; [| by apply IHl].A: Type
P: A → Prop
Pdec: ∀ a : A, Decision (P a)
a: dsig P
l: list (dsig P)
IHl: NoDup (map proj1_sig l) → NoDup l
Ha: `a ∉ map proj1_sig l
H: NoDup (map proj1_sig l)
a ∉ l
    by contradict Ha; apply elem_of_list_fmap; eexists.
Qed.

Lemma is_Some_proj_elim {A : Type}
  (m : option A) (Hsome : is_Some m) (f : A) (H : m = Some f) :
  is_Some_proj Hsome = f.A: Type
m: option A
Hsome: is_Some m
f: A
H: m = Some f
is_Some_proj Hsome = f
Proof.A: Type
m: option A
Hsome: is_Some m
f: A
H: m = Some f
is_Some_proj Hsome = f by intros; subst. Qed.

Definition set_Forall2 `{ElemOf A C} (R : relation A) (X : C) :=
  forall x y : A, x ∈ X -> y ∈ X -> x <> y -> R x y.

Definition set_Exists2 `{ElemOf A C} (R : relation A) (X : C) :=
  exists x y : A, x ∈ X /\ y ∈ X /\ x <> y /\ R x y.

Section sec_Forall2_Exists2_props.

Context
  `{SemiSet A C}
  (R : relation A)
  .

Lemma set_Forall2_empty : set_Forall2 R (∅ : C).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
set_Forall2 R (∅ : C)
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
set_Forall2 R (∅ : C) by unfold set_Forall2; set_solver. Qed.

Lemma set_Forall2_singleton (x : A) : set_Forall2 R ({[ x ]} : C).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x: A
set_Forall2 R ({[x]} : C)
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x: A
set_Forall2 R ({[x]} : C) by unfold set_Forall2; set_solver. Qed.

Lemma set_Forall2_pair (x y : A) :
  x <> y -> set_Forall2 R ({[ x; y ]} : C) <-> R x y /\ R y x.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x, y: A
x ≠ y → set_Forall2 R ({[x; y]} : C) ↔ R x y ∧ R y x
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x, y: A
x ≠ y → set_Forall2 R ({[x; y]} : C) ↔ R x y ∧ R y x by unfold set_Forall2; set_solver. Qed.

Lemma set_Forall2_union (X Y : C) :
  set_Forall2 R X -> set_Forall2 R Y ->
  (forall x y : A, x ∈ X -> y ∈ Y -> x <> y -> R x y /\ R y x) ->
  set_Forall2 R (X ∪ Y).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Forall2 R X
→ set_Forall2 R Y
  → (∀ x y : A, x ∈ X → y ∈ Y → x ≠ y → R x y ∧ R y x)
    → set_Forall2 R (X ∪ Y)
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Forall2 R X
→ set_Forall2 R Y
  → (∀ x y : A, x ∈ X → y ∈ Y → x ≠ y → R x y ∧ R y x)
    → set_Forall2 R (X ∪ Y) by unfold set_Forall2; set_solver. Qed.

Lemma set_Forall2_union_inv_1 (X Y : C) :
  set_Forall2 R (X ∪ Y) -> set_Forall2 R X.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Forall2 R (X ∪ Y) → set_Forall2 R X
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Forall2 R (X ∪ Y) → set_Forall2 R X by unfold set_Forall2; set_solver. Qed.

Lemma set_Forall2_union_inv_2 (X Y : C) :
  set_Forall2 R (X ∪ Y) -> set_Forall2 R Y.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Forall2 R (X ∪ Y) → set_Forall2 R Y
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Forall2 R (X ∪ Y) → set_Forall2 R Y by unfold set_Forall2; set_solver. Qed.

Lemma set_Exists2_empty : ~ set_Exists2 R (∅ : C).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
¬ set_Exists2 R (∅ : C)
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
¬ set_Exists2 R (∅ : C) by unfold set_Exists2; set_solver. Qed.

Lemma set_Exists2_singleton x : ~ set_Exists2 R ({[ x ]} : C).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x: A
¬ set_Exists2 R ({[x]} : C)
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x: A
¬ set_Exists2 R ({[x]} : C) by unfold set_Exists2; set_solver. Qed.

Lemma set_Exists2_pair (x y : A) :
  set_Exists2 R ({[ x; y ]} : C) <-> x <> y /\ (R x y \/ R y x).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x, y: A
set_Exists2 R ({[x; y]} : C) ↔ x ≠ y ∧ (R x y ∨ R y x)
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x, y: A
set_Exists2 R ({[x; y]} : C) ↔ x ≠ y ∧ (R x y ∨ R y x)
  split; [by unfold set_Exists2; set_solver |].A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x, y: A
x ≠ y ∧ (R x y ∨ R y x) → set_Exists2 R {[x; y]}
  intros [? []].A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x, y: A
H4: x ≠ y
H5: R x y
set_Exists2 R {[x; y]}
A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x, y: A
H4: x ≠ y
H5: R y x
set_Exists2 R {[x; y]}
  -A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x, y: A
H4: x ≠ y
H5: R x y
set_Exists2 R {[x; y]} by exists x, y; set_solver.
  -A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
x, y: A
H4: x ≠ y
H5: R y x
set_Exists2 R {[x; y]} by exists y, x; set_solver.
Qed.

Lemma set_Exists2_union_1 (X Y : C) :
  set_Exists2 R X -> set_Exists2 R (X ∪ Y).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Exists2 R X → set_Exists2 R (X ∪ Y)
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Exists2 R X → set_Exists2 R (X ∪ Y) by intros [x [y Hxy]]; exists x, y; set_solver. Qed.

Lemma set_Exists2_union_2 (X Y : C) :
  set_Exists2 R Y -> set_Exists2 R (X ∪ Y).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Exists2 R Y → set_Exists2 R (X ∪ Y)
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Exists2 R Y → set_Exists2 R (X ∪ Y) by intros [x [y Hxy]]; exists x, y; set_solver. Qed.

Lemma set_Exists2_union_3 (X Y : C) (x y : A) :
  x ∈ X -> y ∈ Y -> x <> y -> R x y -> set_Exists2 R (X ∪ Y).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
x ∈ X → y ∈ Y → x ≠ y → R x y → set_Exists2 R (X ∪ Y)
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
x ∈ X → y ∈ Y → x ≠ y → R x y → set_Exists2 R (X ∪ Y) by exists x, y; set_solver. Qed.

Lemma set_Exists2_union_4 (X Y : C) (x y : A) :
  x ∈ X -> y ∈ Y -> x <> y -> R y x -> set_Exists2 R (X ∪ Y).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
x ∈ X → y ∈ Y → x ≠ y → R y x → set_Exists2 R (X ∪ Y)
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
x ∈ X → y ∈ Y → x ≠ y → R y x → set_Exists2 R (X ∪ Y) by intros; exists y, x; set_solver. Qed.

Lemma set_Exists2_union_inv (X Y : C) :
  set_Exists2 R (X ∪ Y) ->
    set_Exists2 R X \/ set_Exists2 R Y \/
    exists x y : A, x ∈ X /\ y ∈ Y /\ x <> y /\ (R x y \/ R y x).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Exists2 R (X ∪ Y)
→ set_Exists2 R X
  ∨ set_Exists2 R Y
    ∨ (∃ x y : A,
         x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x))
Proof.A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
set_Exists2 R (X ∪ Y)
→ set_Exists2 R X
  ∨ set_Exists2 R Y
    ∨ (∃ x y : A,
         x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x))
  intros (x & y & Hx & Hy & Hneq & Hxy).A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
Hx: x ∈ X ∪ Y
Hy: y ∈ X ∪ Y
Hneq: x ≠ y
Hxy: R x y
set_Exists2 R X
∨ set_Exists2 R Y
  ∨ (∃ x y : A,
       x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x))
  apply elem_of_union in Hx as [], Hy as [].A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
H4: x ∈ X
H5: y ∈ X
Hneq: x ≠ y
Hxy: R x y
set_Exists2 R X
∨ set_Exists2 R Y
  ∨ (∃ x y : A,
       x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x))
A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
H4: x ∈ X
H5: y ∈ Y
Hneq: x ≠ y
Hxy: R x y
set_Exists2 R X
∨ set_Exists2 R Y
  ∨ (∃ x y : A,
       x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x))
A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
H4: x ∈ Y
H5: y ∈ X
Hneq: x ≠ y
Hxy: R x y
set_Exists2 R X
∨ set_Exists2 R Y
  ∨ (∃ x y : A,
       x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x))
A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
H4: x ∈ Y
H5: y ∈ Y
Hneq: x ≠ y
Hxy: R x y
set_Exists2 R X
∨ set_Exists2 R Y
  ∨ (∃ x y : A,
       x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x))
  -A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
H4: x ∈ X
H5: y ∈ X
Hneq: x ≠ y
Hxy: R x y
set_Exists2 R X
∨ set_Exists2 R Y
  ∨ (∃ x y : A,
       x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x)) by left; exists x, y.
  -A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
H4: x ∈ X
H5: y ∈ Y
Hneq: x ≠ y
Hxy: R x y
set_Exists2 R X
∨ set_Exists2 R Y
  ∨ (∃ x y : A,
       x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x)) by right; right; exists x, y; repeat split; [.. | left].
  -A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
H4: x ∈ Y
H5: y ∈ X
Hneq: x ≠ y
Hxy: R x y
set_Exists2 R X
∨ set_Exists2 R Y
  ∨ (∃ x y : A,
       x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x)) by right; right; exists y, x; repeat split; [.. | right].
  -A, C: Type
H: ElemOf A C
H0: Empty C
H1: Singleton A C
H2: Union C
H3: SemiSet A C
R: relation A
X, Y: C
x, y: A
H4: x ∈ Y
H5: y ∈ Y
Hneq: x ≠ y
Hxy: R x y
set_Exists2 R X
∨ set_Exists2 R Y
  ∨ (∃ x y : A,
       x ∈ X ∧ y ∈ Y ∧ x ≠ y ∧ (R x y ∨ R y x)) by right; left; exists x, y.
Qed.

End sec_Forall2_Exists2_props.