Cycles of a list

Lists have an equivalence relation of whether they are rotational permutations of one another. This relation is defined as IsRotated.

Based on this, we define the quotient of lists by the rotation relation, called Cycle.

We also define a representation of concrete cycles, available when viewing them in a goal state or via #eval, when over representable types. For example, the cycle (2 1 4 3) will be shown as c[2, 1, 4, 3]. Two equal cycles may be printed differently if their internal representation is different.

def List.nextOr {α : Type u_1} [DecidableEq α] : List α → α → α → α

Return the z such that x :: z :: _ appears in xs, or default if there is no such z.

@[simp]

theorem List.nextOr_nil {α : Type u_1} [DecidableEq α] (x d : α) : [].nextOr x d = d

@[simp]

theorem List.nextOr_singleton {α : Type u_1} [DecidableEq α] (x y d : α) : [y].nextOr x d = d

@[simp]

theorem List.nextOr_self_cons_cons {α : Type u_1} [DecidableEq α] (xs : List α) (x y d : α) : (x :: y :: xs).nextOr x d = y

theorem List.nextOr_cons_of_ne {α : Type u_1} [DecidableEq α] (xs : List α) (y x d : α) (h : x ≠ y) : (y :: xs).nextOr x d = xs.nextOr x d

theorem List.nextOr_eq_nextOr_of_mem_of_ne {α : Type u_1} [DecidableEq α] (xs : List α) (x d d' : α) (x_mem : x ∈ xs) (x_ne : x ≠ xs.getLast ⋯) : xs.nextOr x d = xs.nextOr x d'

nextOr does not depend on the default value, if the next value appears.

theorem List.mem_of_nextOr_ne {α : Type u_1} [DecidableEq α] {xs : List α} {x d : α} (h : xs.nextOr x d ≠ d) : x ∈ xs

theorem List.nextOr_concat {α : Type u_1} [DecidableEq α] {xs : List α} {x : α} (d : α) (h : x ∉ xs) : (xs ++ [x]).nextOr x d = d

theorem List.nextOr_mem {α : Type u_1} [DecidableEq α] {xs : List α} {x d : α} (hd : d ∈ xs) : xs.nextOr x d ∈ xs

def List.next {α : Type u_1} [DecidableEq α] (l : List α) (x : α) (h : x ∈ l) : α

Given an element x : α of l : List α such that x ∈ l, get the next element of l. This works from head to tail, (including a check for last element) so it will match on first hit, ignoring later duplicates.

For example:

• next [1, 2, 3] 2 _ = 3
• next [1, 2, 3] 3 _ = 1
• next [1, 2, 3, 2, 4] 2 _ = 3
• next [1, 2, 3, 2] 2 _ = 3
• next [1, 1, 2, 3, 2] 1 _ = 1

def List.prev {α : Type u_1} [DecidableEq α] (l : List α) (x : α) : x ∈ l → α

Given an element x : α of l : List α such that x ∈ l, get the previous element of l. This works from head to tail, (including a check for last element) so it will match on first hit, ignoring later duplicates.

• prev [1, 2, 3] 2 _ = 1
• prev [1, 2, 3] 1 _ = 3
• prev [1, 2, 3, 2, 4] 2 _ = 1
• prev [1, 2, 3, 4, 2] 2 _ = 1
• prev [1, 1, 2] 1 _ = 2

@[simp]

theorem List.next_singleton {α : Type u_1} [DecidableEq α] (x y : α) (h : x ∈ [y]) : [y].next x h = y

@[simp]

theorem List.prev_singleton {α : Type u_1} [DecidableEq α] (x y : α) (h : x ∈ [y]) : [y].prev x h = y

theorem List.next_cons_cons_eq' {α : Type u_1} [DecidableEq α] (l : List α) (x y z : α) (h : x ∈ y :: z :: l) (hx : x = y) : (y :: z :: l).next x h = z

@[simp]

theorem List.next_cons_cons_eq {α : Type u_1} [DecidableEq α] (l : List α) (x z : α) (h : x ∈ x :: z :: l) : (x :: z :: l).next x h = z

theorem List.next_ne_head_ne_getLast {α : Type u_1} [DecidableEq α] (l : List α) (x : α) (h : x ∈ l) (y : α) (h✝ : x ∈ y :: l) (hy : x ≠ y) (hx : x ≠ (y :: l).getLast ⋯) : (y :: l).next x h✝ = l.next x ⋯

theorem List.next_cons_concat {α : Type u_1} [DecidableEq α] (l : List α) (x y : α) (hy : x ≠ y) (hx : x ∉ l) (h : x ∈ y :: l ++ [x] := ⋯) : (y :: l ++ [x]).next x h = y

theorem List.next_getLast_cons {α : Type u_1} [DecidableEq α] (l : List α) (x : α) (h : x ∈ l) (y : α) (h✝ : x ∈ y :: l) (hy : x ≠ y) (hx : x = (y :: l).getLast ⋯) (hl : l.Nodup) : (y :: l).next x h✝ = y

theorem List.prev_getLast_cons' {α : Type u_1} [DecidableEq α] (l : List α) (x y : α) (hxy : x ∈ y :: l) (hx : x = y) : (y :: l).prev x hxy = (y :: l).getLast ⋯

@[simp]

theorem List.prev_getLast_cons {α : Type u_1} [DecidableEq α] (l : List α) (x : α) (h : x ∈ x :: l) : (x :: l).prev x h = (x :: l).getLast ⋯

theorem List.prev_cons_cons_eq' {α : Type u_1} [DecidableEq α] (l : List α) (x y z : α) (h : x ∈ y :: z :: l) (hx : x = y) : (y :: z :: l).prev x h = (z :: l).getLast ⋯

theorem List.prev_cons_cons_eq {α : Type u_1} [DecidableEq α] (l : List α) (x z : α) (h : x ∈ x :: z :: l) : (x :: z :: l).prev x h = (z :: l).getLast ⋯

theorem List.prev_cons_cons_of_ne' {α : Type u_1} [DecidableEq α] (l : List α) (x y z : α) (h : x ∈ y :: z :: l) (hy : x ≠ y) (hz : x = z) : (y :: z :: l).prev x h = y

theorem List.prev_cons_cons_of_ne {α : Type u_1} [DecidableEq α] (l : List α) (x y : α) (h : x ∈ y :: x :: l) (hy : x ≠ y) : (y :: x :: l).prev x h = y

theorem List.prev_ne_cons_cons {α : Type u_1} [DecidableEq α] (l : List α) (x y z : α) (h : x ∈ y :: z :: l) (hy : x ≠ y) (hz : x ≠ z) : (y :: z :: l).prev x h = (z :: l).prev x ⋯

theorem List.next_mem {α : Type u_1} [DecidableEq α] (l : List α) (x : α) (h : x ∈ l) : l.next x h ∈ l

theorem List.prev_mem {α : Type u_1} [DecidableEq α] (l : List α) (x : α) (h : x ∈ l) : l.prev x h ∈ l

theorem List.next_getElem {α : Type u_1} [DecidableEq α] (l : List α) (h : l.Nodup) (i : ℕ) (hi : i < l.length) : l.next l[i] ⋯ = l[(i + 1) % l.length]

@[deprecated List.next_getElem (since := "2025-02-15")]

theorem List.next_get {α : Type u_1} [DecidableEq α] (l : List α) (h : l.Nodup) (i : ℕ) (hi : i < l.length) : l.next l[i] ⋯ = l[(i + 1) % l.length]

Alias of List.next_getElem.

theorem List.prev_getElem {α : Type u_1} [DecidableEq α] (l : List α) (h : l.Nodup) (i : ℕ) (hi : i < l.length) : l.prev l[i] ⋯ = l[(i + (l.length - 1)) % l.length]

@[deprecated List.prev_getElem (since := "2025-02-15")]

theorem List.prev_get {α : Type u_1} [DecidableEq α] (l : List α) (h : l.Nodup) (i : ℕ) (hi : i < l.length) : l.prev l[i] ⋯ = l[(i + (l.length - 1)) % l.length]

Alias of List.prev_getElem.

theorem List.pmap_next_eq_rotate_one {α : Type u_1} [DecidableEq α] (l : List α) (h : l.Nodup) : pmap l.next l ⋯ = l.rotate 1

theorem List.pmap_prev_eq_rotate_length_sub_one {α : Type u_1} [DecidableEq α] (l : List α) (h : l.Nodup) : pmap l.prev l ⋯ = l.rotate (l.length - 1)

theorem List.prev_next {α : Type u_1} [DecidableEq α] (l : List α) (h : l.Nodup) (x : α) (hx : x ∈ l) : l.prev (l.next x hx) ⋯ = x

theorem List.next_prev {α : Type u_1} [DecidableEq α] (l : List α) (h : l.Nodup) (x : α) (hx : x ∈ l) : l.next (l.prev x hx) ⋯ = x

theorem List.prev_reverse_eq_next {α : Type u_1} [DecidableEq α] (l : List α) (h : l.Nodup) (x : α) (hx : x ∈ l) : l.reverse.prev x ⋯ = l.next x hx

theorem List.next_reverse_eq_prev {α : Type u_1} [DecidableEq α] (l : List α) (h : l.Nodup) (x : α) (hx : x ∈ l) : l.reverse.next x ⋯ = l.prev x hx

theorem List.isRotated_next_eq {α : Type u_1} [DecidableEq α] {l l' : List α} (h : l ~r l') (hn : l.Nodup) {x : α} (hx : x ∈ l) : l.next x hx = l'.next x ⋯

theorem List.isRotated_prev_eq {α : Type u_1} [DecidableEq α] {l l' : List α} (h : l ~r l') (hn : l.Nodup) {x : α} (hx : x ∈ l) : l.prev x hx = l'.prev x ⋯

def Cycle (α : Type u_1) : Type u_1

Cycle α is the quotient of List α by cyclic permutation. Duplicates are allowed.

def Cycle.ofList {α : Type u_1} : List α → Cycle α

The coercion from List α to Cycle α

instance Cycle.instCoeList {α : Type u_1} : Coe (List α) (Cycle α)

@[simp]

theorem Cycle.coe_eq_coe {α : Type u_1} {l₁ l₂ : List α} : ↑l₁ = ↑l₂ ↔ l₁ ~r l₂

@[simp]

theorem Cycle.mk_eq_coe {α : Type u_1} (l : List α) : Quot.mk (⇑(List.IsRotated.setoid α)) l = ↑l

@[simp]

theorem Cycle.mk''_eq_coe {α : Type u_1} (l : List α) : Quotient.mk'' l = ↑l

theorem Cycle.coe_cons_eq_coe_append {α : Type u_1} (l : List α) (a : α) : ↑(a :: l) = ↑(l ++ [a])

def Cycle.nil {α : Type u_1} : Cycle α

The unique empty cycle.

@[simp]

theorem Cycle.coe_nil {α : Type u_1} : ↑[] = nil

@[simp]

theorem Cycle.coe_eq_nil {α : Type u_1} (l : List α) : ↑l = nil ↔ l = []

instance Cycle.instEmptyCollection {α : Type u_1} : EmptyCollection (Cycle α)

For consistency with EmptyCollection (List α).

@[simp]

theorem Cycle.empty_eq {α : Type u_1} : ∅ = nil

instance Cycle.instInhabited {α : Type u_1} : Inhabited (Cycle α)

theorem Cycle.induction_on {α : Type u_1} {C : Cycle α → Prop} (s : Cycle α) (H0 : C nil) (HI : ∀ (a : α) (l : List α), C ↑l → C ↑(a :: l)) : C s

An induction principle for Cycle. Use as induction s.

def Cycle.Mem {α : Type u_1} (s : Cycle α) (a : α) : Prop

For x : α, s : Cycle α, x ∈ s indicates that x occurs at least once in s.

instance Cycle.instMembership {α : Type u_1} : Membership α (Cycle α)

@[simp]

theorem Cycle.mem_coe_iff {α : Type u_1} {a : α} {l : List α} : a ∈ ↑l ↔ a ∈ l

@[simp]

theorem Cycle.notMem_nil {α : Type u_1} (a : α) : a ∉ nil

@[deprecated Cycle.notMem_nil (since := "2025-05-23")]

theorem Cycle.not_mem_nil {α : Type u_1} (a : α) : a ∉ nil

Alias of Cycle.notMem_nil.

instance Cycle.instDecidableEq {α : Type u_1} [DecidableEq α] : DecidableEq (Cycle α)

instance Cycle.instDecidableMemOfDecidableEq {α : Type u_1} [DecidableEq α] (x : α) (s : Cycle α) : Decidable (x ∈ s)

def Cycle.reverse {α : Type u_1} (s : Cycle α) : Cycle α

Reverse a s : Cycle α by reversing the underlying List.

@[simp]

theorem Cycle.reverse_coe {α : Type u_1} (l : List α) : (↑l).reverse = ↑l.reverse

@[simp]

theorem Cycle.mem_reverse_iff {α : Type u_1} {a : α} {s : Cycle α} : a ∈ s.reverse ↔ a ∈ s

@[simp]

theorem Cycle.reverse_reverse {α : Type u_1} (s : Cycle α) : s.reverse.reverse = s

@[simp]

theorem Cycle.reverse_nil {α : Type u_1} : nil.reverse = nil

def Cycle.length {α : Type u_1} (s : Cycle α) : ℕ

The length of the s : Cycle α, which is the number of elements, counting duplicates.

@[simp]

theorem Cycle.length_coe {α : Type u_1} (l : List α) : (↑l).length = l.length

@[simp]

theorem Cycle.length_nil {α : Type u_1} : nil.length = 0

@[simp]

theorem Cycle.length_reverse {α : Type u_1} (s : Cycle α) : s.reverse.length = s.length

def Cycle.Subsingleton {α : Type u_1} (s : Cycle α) : Prop

A s : Cycle α that is at most one element.

theorem Cycle.subsingleton_nil {α : Type u_1} : nil.Subsingleton

theorem Cycle.length_subsingleton_iff {α : Type u_1} {s : Cycle α} : s.Subsingleton ↔ s.length ≤ 1

@[simp]

theorem Cycle.subsingleton_reverse_iff {α : Type u_1} {s : Cycle α} : s.reverse.Subsingleton ↔ s.Subsingleton

theorem Cycle.Subsingleton.congr {α : Type u_1} {s : Cycle α} (h : s.Subsingleton) ⦃x : α⦄ (_hx : x ∈ s) ⦃y : α⦄ (_hy : y ∈ s) : x = y

def Cycle.Nontrivial {α : Type u_1} (s : Cycle α) : Prop

A s : Cycle α that is made up of at least two unique elements.

@[simp]

theorem Cycle.nontrivial_coe_nodup_iff {α : Type u_1} {l : List α} (hl : l.Nodup) : (↑l).Nontrivial ↔ 2 ≤ l.length

@[simp]

theorem Cycle.nontrivial_reverse_iff {α : Type u_1} {s : Cycle α} : s.reverse.Nontrivial ↔ s.Nontrivial

theorem Cycle.length_nontrivial {α : Type u_1} {s : Cycle α} (h : s.Nontrivial) : 2 ≤ s.length

def Cycle.Nodup {α : Type u_1} (s : Cycle α) : Prop

The s : Cycle α contains no duplicates.

@[simp]

theorem Cycle.nodup_nil {α : Type u_1} : nil.Nodup

@[simp]

theorem Cycle.nodup_coe_iff {α : Type u_1} {l : List α} : (↑l).Nodup ↔ l.Nodup

@[simp]

theorem Cycle.nodup_reverse_iff {α : Type u_1} {s : Cycle α} : s.reverse.Nodup ↔ s.Nodup

theorem Cycle.Subsingleton.nodup {α : Type u_1} {s : Cycle α} (h : s.Subsingleton) : s.Nodup

theorem Cycle.Nodup.nontrivial_iff {α : Type u_1} {s : Cycle α} (h : s.Nodup) : s.Nontrivial ↔ ¬s.Subsingleton

def Cycle.toMultiset {α : Type u_1} (s : Cycle α) : Multiset α

The s : Cycle α as a Multiset α.

@[simp]

theorem Cycle.coe_toMultiset {α : Type u_1} (l : List α) : (↑l).toMultiset = ↑l

@[simp]

theorem Cycle.nil_toMultiset {α : Type u_1} : nil.toMultiset = 0

@[simp]

theorem Cycle.card_toMultiset {α : Type u_1} (s : Cycle α) : s.toMultiset.card = s.length

@[simp]

theorem Cycle.toMultiset_eq_nil {α : Type u_1} {s : Cycle α} : s.toMultiset = 0 ↔ s = nil

def Cycle.map {α : Type u_1} {β : Type u_2} (f : α → β) : Cycle α → Cycle β

The lift of list.map.

@[simp]

theorem Cycle.map_nil {α : Type u_1} {β : Type u_2} (f : α → β) : map f nil = nil

@[simp]

theorem Cycle.map_coe {α : Type u_1} {β : Type u_2} (f : α → β) (l : List α) : map f ↑l = ↑(List.map f l)

@[simp]

theorem Cycle.map_eq_nil {α : Type u_1} {β : Type u_2} (f : α → β) (s : Cycle α) : map f s = nil ↔ s = nil

@[simp]

theorem Cycle.mem_map {α : Type u_1} {β : Type u_2} {f : α → β} {b : β} {s : Cycle α} : b ∈ map f s ↔ ∃ a ∈ s, f a = b

def Cycle.lists {α : Type u_1} (s : Cycle α) : Multiset (List α)

The Multiset of lists that can make the cycle.

@[simp]

theorem Cycle.lists_coe {α : Type u_1} (l : List α) : (↑l).lists = ↑l.cyclicPermutations

@[simp]

theorem Cycle.mem_lists_iff_coe_eq {α : Type u_1} {s : Cycle α} {l : List α} : l ∈ s.lists ↔ ↑l = s

@[simp]

theorem Cycle.lists_nil {α : Type u_1} : nil.lists = {[]}

@[irreducible]

def Cycle.decidableNontrivialCoe {α : Type u_1} [DecidableEq α] (l : List α) : Decidable (↑l).Nontrivial

Auxiliary decidability algorithm for lists that contain at least two unique elements.

instance Cycle.instDecidableNontrivial {α : Type u_1} [DecidableEq α] {s : Cycle α} : Decidable s.Nontrivial

instance Cycle.instDecidableNodup {α : Type u_1} [DecidableEq α] {s : Cycle α} : Decidable s.Nodup

instance Cycle.fintypeNodupCycle {α : Type u_1} [DecidableEq α] [Fintype α] : Fintype { s : Cycle α // s.Nodup }

instance Cycle.fintypeNodupNontrivialCycle {α : Type u_1} [DecidableEq α] [Fintype α] : Fintype { s : Cycle α // s.Nodup ∧ s.Nontrivial }

def Cycle.toFinset {α : Type u_1} [DecidableEq α] (s : Cycle α) : Finset α

The s : Cycle α as a Finset α.

@[simp]

theorem Cycle.toFinset_toMultiset {α : Type u_1} [DecidableEq α] (s : Cycle α) : s.toMultiset.toFinset = s.toFinset

@[simp]

theorem Cycle.coe_toFinset {α : Type u_1} [DecidableEq α] (l : List α) : (↑l).toFinset = l.toFinset

@[simp]

theorem Cycle.nil_toFinset {α : Type u_1} [DecidableEq α] : nil.toFinset = ∅

@[simp]

theorem Cycle.toFinset_eq_nil {α : Type u_1} [DecidableEq α] {s : Cycle α} : s.toFinset = ∅ ↔ s = nil

def Cycle.next {α : Type u_1} [DecidableEq α] (s : Cycle α) (_hs : s.Nodup) (x : α) (_hx : x ∈ s) : α

Given a s : Cycle α such that Nodup s, retrieve the next element after x ∈ s.

def Cycle.prev {α : Type u_1} [DecidableEq α] (s : Cycle α) (_hs : s.Nodup) (x : α) (_hx : x ∈ s) : α

Given a s : Cycle α such that Nodup s, retrieve the previous element before x ∈ s.

theorem Cycle.prev_reverse_eq_next {α : Type u_1} [DecidableEq α] (s : Cycle α) (hs : s.Nodup) (x : α) (hx : x ∈ s) : s.reverse.prev ⋯ x ⋯ = s.next hs x hx

@[simp]

theorem Cycle.prev_reverse_eq_next' {α : Type u_1} [DecidableEq α] (s : Cycle α) (hs : s.reverse.Nodup) (x : α) (hx : x ∈ s.reverse) : s.reverse.prev hs x hx = s.next ⋯ x ⋯

theorem Cycle.next_reverse_eq_prev {α : Type u_1} [DecidableEq α] (s : Cycle α) (hs : s.Nodup) (x : α) (hx : x ∈ s) : s.reverse.next ⋯ x ⋯ = s.prev hs x hx

@[simp]

theorem Cycle.next_reverse_eq_prev' {α : Type u_1} [DecidableEq α] (s : Cycle α) (hs : s.reverse.Nodup) (x : α) (hx : x ∈ s.reverse) : s.reverse.next hs x hx = s.prev ⋯ x ⋯

@[simp]

theorem Cycle.next_mem {α : Type u_1} [DecidableEq α] (s : Cycle α) (hs : s.Nodup) (x : α) (hx : x ∈ s) : s.next hs x hx ∈ s

theorem Cycle.prev_mem {α : Type u_1} [DecidableEq α] (s : Cycle α) (hs : s.Nodup) (x : α) (hx : x ∈ s) : s.prev hs x hx ∈ s

@[simp]

theorem Cycle.prev_next {α : Type u_1} [DecidableEq α] (s : Cycle α) (hs : s.Nodup) (x : α) (hx : x ∈ s) : s.prev hs (s.next hs x hx) ⋯ = x

@[simp]

theorem Cycle.next_prev {α : Type u_1} [DecidableEq α] (s : Cycle α) (hs : s.Nodup) (x : α) (hx : x ∈ s) : s.next hs (s.prev hs x hx) ⋯ = x

unsafe instance Cycle.instRepr {α : Type u_1} [Repr α] : Repr (Cycle α)

We define a representation of concrete cycles, available when viewing them in a goal state or via #eval, when over representable types. For example, the cycle (2 1 4 3) will be shown as c[2, 1, 4, 3]. Two equal cycles may be printed differently if their internal representation is different.

def Cycle.Chain {α : Type u_1} (r : α → α → Prop) (c : Cycle α) : Prop

chain R s means that R holds between adjacent elements of s.

chain R ([a, b, c] : Cycle α) ↔ R a b ∧ R b c ∧ R c a

@[simp]

theorem Cycle.Chain.nil {α : Type u_1} (r : α → α → Prop) : Chain r Cycle.nil

@[simp]

theorem Cycle.chain_coe_cons {α : Type u_1} (r : α → α → Prop) (a : α) (l : List α) : Chain r ↑(a :: l) ↔ List.Chain r a (l ++ [a])

theorem Cycle.chain_singleton {α : Type u_1} (r : α → α → Prop) (a : α) : Chain r ↑[a] ↔ r a a

theorem Cycle.chain_ne_nil {α : Type u_1} (r : α → α → Prop) {l : List α} (hl : l ≠ []) : Chain r ↑l ↔ List.Chain r (l.getLast hl) l

theorem Cycle.chain_map {α : Type u_1} {β : Type u_2} {r : α → α → Prop} (f : β → α) {s : Cycle β} : Chain r (map f s) ↔ Chain (fun (a b : β) => r (f a) (f b)) s

theorem Cycle.chain_range_succ (r : ℕ → ℕ → Prop) (n : ℕ) : Chain r ↑(List.range n.succ) ↔ r n 0 ∧ ∀ m < n, r m m.succ

theorem Cycle.Chain.imp {α : Type u_1} {s : Cycle α} {r₁ r₂ : α → α → Prop} (H : ∀ (a b : α), r₁ a b → r₂ a b) (p : Chain r₁ s) : Chain r₂ s

theorem Cycle.chain_mono {α : Type u_1} : Monotone Chain

As a function from a relation to a predicate, chain is monotonic.

theorem Cycle.chain_of_pairwise {α : Type u_1} {r : α → α → Prop} {s : Cycle α} : (∀ a ∈ s, ∀ b ∈ s, r a b) → Chain r s

theorem Cycle.chain_iff_pairwise {α : Type u_1} {r : α → α → Prop} {s : Cycle α} [IsTrans α r] : Chain r s ↔ ∀ a ∈ s, ∀ b ∈ s, r a b

theorem Cycle.Chain.eq_nil_of_irrefl {α : Type u_1} {r : α → α → Prop} {s : Cycle α} [IsTrans α r] [IsIrrefl α r] (h : Chain r s) : s = Cycle.nil

theorem Cycle.Chain.eq_nil_of_well_founded {α : Type u_1} {r : α → α → Prop} {s : Cycle α} [IsWellFounded α r] (h : Chain r s) : s = Cycle.nil

theorem Cycle.forall_eq_of_chain {α : Type u_1} {r : α → α → Prop} {s : Cycle α} [IsTrans α r] [IsAntisymm α r] (hs : Chain r s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : a = b