# Documentation

Mathlib.Data.List.Forall2

# Double universal quantification on a list #

This file provides an API for List.Forall₂ (definition in Data.List.Defs). Forall₂ R l₁ l₂ means that l₁ and l₂ have the same length, and whenever a is the nth element of l₁, and b is the nth element of l₂, then R a b is satisfied.

theorem List.forall₂_iff {α : Type u_1} {β : Type u_2} (R : αβProp) :
∀ (a : List α) (a_1 : List β), List.Forall₂ R a a_1 a = [] a_1 = [] Exists fun {a_2} => Exists fun {b} => Exists fun {l₁} => Exists fun {l₂} => R a_2 b List.Forall₂ R l₁ l₂ a = a_2 :: l₁ a_1 = b :: l₂
@[simp]
theorem List.forall₂_cons {α : Type u_1} {β : Type u_2} {R : αβProp} {a : α} {b : β} {l₁ : List α} {l₂ : List β} :
List.Forall₂ R (a :: l₁) (b :: l₂) R a b List.Forall₂ R l₁ l₂
theorem List.Forall₂.imp {α : Type u_1} {β : Type u_2} {R : αβProp} {S : αβProp} (H : (a : α) → (b : β) → R a bS a b) {l₁ : List α} {l₂ : List β} (h : List.Forall₂ R l₁ l₂) :
List.Forall₂ S l₁ l₂
theorem List.Forall₂.mp {α : Type u_1} {β : Type u_2} {R : αβProp} {S : αβProp} {Q : αβProp} (h : (a : α) → (b : β) → Q a bR a bS a b) {l₁ : List α} {l₂ : List β} :
List.Forall₂ Q l₁ l₂List.Forall₂ R l₁ l₂List.Forall₂ S l₁ l₂
theorem List.Forall₂.flip {α : Type u_1} {β : Type u_2} {R : αβProp} {a : List α} {b : List β} :
@[simp]
theorem List.forall₂_same {α : Type u_1} {Rₐ : ααProp} {l : List α} :
List.Forall₂ Rₐ l l (x : α) → x lRₐ x x
theorem List.forall₂_refl {α : Type u_1} {Rₐ : ααProp} [inst : IsRefl α Rₐ] (l : List α) :
@[simp]
theorem List.forall₂_eq_eq_eq {α : Type u_1} :
(List.Forall₂ fun x x_1 => x = x_1) = Eq
@[simp]
theorem List.forall₂_nil_left_iff {α : Type u_2} {β : Type u_1} {R : αβProp} {l : List β} :
List.Forall₂ R [] l l = []
@[simp]
theorem List.forall₂_nil_right_iff {α : Type u_1} {β : Type u_2} {R : αβProp} {l : List α} :
List.Forall₂ R l [] l = []
theorem List.forall₂_cons_left_iff {α : Type u_1} {β : Type u_2} {R : αβProp} {a : α} {l : List α} {u : List β} :
List.Forall₂ R (a :: l) u b u', R a b List.Forall₂ R l u' u = b :: u'
theorem List.forall₂_cons_right_iff {α : Type u_2} {β : Type u_1} {R : αβProp} {b : β} {l : List β} {u : List α} :
List.Forall₂ R u (b :: l) a u', R a b List.Forall₂ R u' l u = a :: u'
theorem List.forall₂_and_left {α : Type u_1} {β : Type u_2} {R : αβProp} {p : αProp} (l : List α) (u : List β) :
List.Forall₂ (fun a b => p a R a b) l u ((a : α) → a lp a) List.Forall₂ R l u
@[simp]
theorem List.forall₂_map_left_iff {α : Type u_3} {β : Type u_2} {γ : Type u_1} {R : αβProp} {f : γα} {l : List γ} {u : List β} :
List.Forall₂ R (List.map f l) u List.Forall₂ (fun c b => R (f c) b) l u
@[simp]
theorem List.forall₂_map_right_iff {α : Type u_1} {β : Type u_3} {γ : Type u_2} {R : αβProp} {f : γβ} {l : List α} {u : List γ} :
List.Forall₂ R l (List.map f u) List.Forall₂ (fun a c => R a (f c)) l u
theorem List.left_unique_forall₂' {α : Type u_1} {β : Type u_2} {R : αβProp} (hr : ) {a : List α} {b : List α} {c : List β} :
List.Forall₂ R a cList.Forall₂ R b ca = b
theorem Relator.LeftUnique.forall₂ {α : Type u_1} {β : Type u_2} {R : αβProp} (hr : ) :
theorem List.right_unique_forall₂' {α : Type u_1} {β : Type u_2} {R : αβProp} (hr : ) {a : List α} {b : List β} {c : List β} :
List.Forall₂ R a bList.Forall₂ R a cb = c
theorem Relator.RightUnique.forall₂ {α : Type u_1} {β : Type u_2} {R : αβProp} (hr : ) :
theorem Relator.BiUnique.forall₂ {α : Type u_1} {β : Type u_2} {R : αβProp} (hr : ) :
theorem List.Forall₂.length_eq {α : Type u_1} {β : Type u_2} {R : αβProp} {l₁ : List α} {l₂ : List β} :
List.Forall₂ R l₁ l₂ =
theorem List.Forall₂.nthLe {α : Type u_1} {β : Type u_2} {R : αβProp} {x : List α} {y : List β} :
List.Forall₂ R x yi : ⦄ → (hx : i < ) → (hy : i < ) → R (List.nthLe x i hx) (List.nthLe y i hy)
theorem List.forall₂_of_length_eq_of_nthLe {α : Type u_1} {β : Type u_2} {R : αβProp} {x : List α} {y : List β} :
((i : ) → (h₁ : i < ) → (h₂ : i < ) → R (List.nthLe x i h₁) (List.nthLe y i h₂)) → List.Forall₂ R x y
theorem List.forall₂_iff_nthLe {α : Type u_1} {β : Type u_2} {R : αβProp} {l₁ : List α} {l₂ : List β} :
List.Forall₂ R l₁ l₂ = ((i : ) → (h₁ : i < ) → (h₂ : i < ) → R (List.nthLe l₁ i h₁) (List.nthLe l₂ i h₂))
theorem List.forall₂_zip {α : Type u_1} {β : Type u_2} {R : αβProp} {l₁ : List α} {l₂ : List β} :
List.Forall₂ R l₁ l₂{a : α} → {b : β} → (a, b) List.zip l₁ l₂R a b
theorem List.forall₂_iff_zip {α : Type u_1} {β : Type u_2} {R : αβProp} {l₁ : List α} {l₂ : List β} :
List.Forall₂ R l₁ l₂ = ({a : α} → {b : β} → (a, b) List.zip l₁ l₂R a b)
theorem List.forall₂_take {α : Type u_1} {β : Type u_2} {R : αβProp} (n : ) {l₁ : List α} {l₂ : List β} :
List.Forall₂ R l₁ l₂List.Forall₂ R (List.take n l₁) (List.take n l₂)
theorem List.forall₂_drop {α : Type u_1} {β : Type u_2} {R : αβProp} (n : ) {l₁ : List α} {l₂ : List β} :
List.Forall₂ R l₁ l₂List.Forall₂ R (List.drop n l₁) (List.drop n l₂)
theorem List.forall₂_take_append {α : Type u_1} {β : Type u_2} {R : αβProp} (l : List α) (l₁ : List β) (l₂ : List β) (h : List.Forall₂ R l (l₁ ++ l₂)) :
theorem List.forall₂_drop_append {α : Type u_1} {β : Type u_2} {R : αβProp} (l : List α) (l₁ : List β) (l₂ : List β) (h : List.Forall₂ R l (l₁ ++ l₂)) :
theorem List.rel_mem {α : Type u_1} {β : Type u_2} {R : αβProp} (hr : ) :
(R ) (fun x x_1 => x x_1) fun x x_1 => x x_1
theorem List.rel_map {α : Type u_1} {β : Type u_3} {γ : Type u_2} {δ : Type u_4} {R : αβProp} {P : γδProp} :
((R P) ) List.map List.map
theorem List.rel_append {α : Type u_1} {β : Type u_2} {R : αβProp} :
) (fun x x_1 => x ++ x_1) fun x x_1 => x ++ x_1
theorem List.rel_reverse {α : Type u_1} {β : Type u_2} {R : αβProp} :
) List.reverse List.reverse
@[simp]
theorem List.forall₂_reverse_iff {α : Type u_1} {β : Type u_2} {R : αβProp} {l₁ : List α} {l₂ : List β} :
List.Forall₂ R () () List.Forall₂ R l₁ l₂
theorem List.rel_join {α : Type u_1} {β : Type u_2} {R : αβProp} :
) List.join List.join
theorem List.rel_bind {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {R : αβProp} {P : γδProp} :
() ) List.bind List.bind
theorem List.rel_foldl {α : Type u_1} {β : Type u_3} {γ : Type u_2} {δ : Type u_4} {R : αβProp} {P : γδProp} :
((P R P) P ) List.foldl List.foldl
theorem List.rel_foldr {α : Type u_1} {β : Type u_3} {γ : Type u_2} {δ : Type u_4} {R : αβProp} {P : γδProp} :
((R P P) P ) List.foldr List.foldr
theorem List.rel_filter {α : Type u_1} {β : Type u_2} {R : αβProp} {p : αBool} {q : βBool} (hpq : (R fun x x_1 => x x_1) (fun x => p x = true) fun x => q x = true) :
) () ()
theorem List.rel_filterMap {α : Type u_1} {β : Type u_3} {γ : Type u_2} {δ : Type u_4} {R : αβProp} {P : γδProp} :
) ) List.filterMap List.filterMap
theorem List.rel_sum {α : Type u_1} {β : Type u_2} {R : αβProp} [inst : ] [inst : ] (h : R 0 0) (hf : (R R R) (fun x x_1 => x + x_1) fun x x_1 => x + x_1) :
R) List.sum List.sum
theorem List.rel_prod {α : Type u_1} {β : Type u_2} {R : αβProp} [inst : ] [inst : ] (h : R 1 1) (hf : (R R R) (fun x x_1 => x * x_1) fun x x_1 => x * x_1) :
R) List.prod List.prod
inductive List.SublistForall₂ {α : Type u_1} {β : Type u_2} (R : αβProp) :
List αList βProp

Given a relation R, sublist_forall₂ r l₁ l₂ indicates that there is a sublist of l₂ such that forall₂ r l₁ l₂.

Instances For
theorem List.sublistForall₂_iff {α : Type u_1} {β : Type u_2} {R : αβProp} {l₁ : List α} {l₂ : List β} :
List.SublistForall₂ R l₁ l₂ l, List.Forall₂ R l₁ l List.Sublist l l₂
instance List.SublistForall₂.is_refl {α : Type u_1} {Rₐ : ααProp} [inst : IsRefl α Rₐ] :
IsRefl (List α) ()
Equations
instance List.SublistForall₂.is_trans {α : Type u_1} {Rₐ : ααProp} [inst : IsTrans α Rₐ] :
IsTrans (List α) ()
Equations
theorem List.Sublist.sublistForall₂ {α : Type u_1} {Rₐ : ααProp} {l₁ : List α} {l₂ : List α} (h : List.Sublist l₁ l₂) [inst : IsRefl α Rₐ] :
List.SublistForall₂ Rₐ l₁ l₂
theorem List.tail_sublistForall₂_self {α : Type u_1} {Rₐ : ααProp} [inst : IsRefl α Rₐ] (l : List α) :