Documentation

Mathlib.Data.Vector3

Alternate definition of Vector in terms of Fin2 #

This file provides a locale Vector3 which overrides the [a, b, c] notation to create a Vector3 instead of a List.

The :: notation is also overloaded by this file to mean Vector3.cons.

def Vector3 (α : Type u) (n : ) :

Alternate definition of Vector based on Fin2.

Equations
Instances For
    instance instInhabitedVector3 {α : Type u_1} {n : } [Inhabited α] :
    Equations
    • instInhabitedVector3 = { default := fun (x : Fin2 n) => default }
    @[match_pattern]
    def Vector3.nil {α : Type u_1} :
    Vector3 α 0

    The empty vector

    Equations
    • [] = nomatch a
    Instances For
      @[match_pattern]
      def Vector3.cons {α : Type u_1} {n : } (a : α) (v : Vector3 α n) :
      Vector3 α (n + 1)

      The vector cons operation

      Equations
      Instances For
        Equations
        • One or more equations did not get rendered due to their size.
        Instances For
          Equations
          • One or more equations did not get rendered due to their size.
          Instances For

            The vector cons operation

            Equations
            Instances For
              @[simp]
              theorem Vector3.cons_fz {α : Type u_1} {n : } (a : α) (v : Vector3 α n) :
              Vector3.cons a v Fin2.fz = a
              @[simp]
              theorem Vector3.cons_fs {α : Type u_1} {n : } (a : α) (v : Vector3 α n) (i : Fin2 n) :
              Vector3.cons a v i.fs = v i
              @[reducible, inline]
              abbrev Vector3.nth {α : Type u_1} {n : } (i : Fin2 n) (v : Vector3 α n) :
              α

              Get the ith element of a vector

              Equations
              Instances For
                @[reducible, inline]
                abbrev Vector3.ofFn {α : Type u_1} {n : } (f : Fin2 nα) :
                Vector3 α n

                Construct a vector from a function on Fin2.

                Equations
                Instances For
                  def Vector3.head {α : Type u_1} {n : } (v : Vector3 α (n + 1)) :
                  α

                  Get the head of a nonempty vector.

                  Equations
                  • v.head = v Fin2.fz
                  Instances For
                    def Vector3.tail {α : Type u_1} {n : } (v : Vector3 α (n + 1)) :
                    Vector3 α n

                    Get the tail of a nonempty vector.

                    Equations
                    • v.tail i = v i.fs
                    Instances For
                      theorem Vector3.eq_nil {α : Type u_1} (v : Vector3 α 0) :
                      v = []
                      theorem Vector3.cons_head_tail {α : Type u_1} {n : } (v : Vector3 α (n + 1)) :
                      Vector3.cons v.head v.tail = v
                      def Vector3.nilElim {α : Type u_1} {C : Vector3 α 0Sort u} (H : C []) (v : Vector3 α 0) :
                      C v

                      Eliminator for an empty vector.

                      Equations
                      Instances For
                        def Vector3.consElim {α : Type u_1} {n : } {C : Vector3 α (n + 1)Sort u} (H : (a : α) → (t : Vector3 α n) → C (Vector3.cons a t)) (v : Vector3 α (n + 1)) :
                        C v

                        Recursion principle for a nonempty vector.

                        Equations
                        Instances For
                          @[simp]
                          theorem Vector3.consElim_cons {α : Type u_1} {n : } {C : Vector3 α (n + 1)Sort u_2} {H : (a : α) → (t : Vector3 α n) → C (Vector3.cons a t)} {a : α} {t : Vector3 α n} :
                          def Vector3.recOn {α : Type u_1} {C : {n : } → Vector3 α nSort u} {n : } (v : Vector3 α n) (H0 : C []) (Hs : {n : } → (a : α) → (w : Vector3 α n) → C wC (Vector3.cons a w)) :
                          C v

                          Recursion principle with the vector as first argument.

                          Equations
                          Instances For
                            @[simp]
                            theorem Vector3.recOn_nil {α : Type u_1} {C : {n : } → Vector3 α nSort u_2} {H0 : C []} {Hs : {n : } → (a : α) → (w : Vector3 α n) → C wC (Vector3.cons a w)} :
                            [].recOn H0 Hs = H0
                            @[simp]
                            theorem Vector3.recOn_cons {α : Type u_1} {C : {n : } → Vector3 α nSort u_2} {H0 : C []} {Hs : {n : } → (a : α) → (w : Vector3 α n) → C wC (Vector3.cons a w)} {n : } {a : α} {v : Vector3 α n} :
                            (Vector3.cons a v).recOn H0 Hs = Hs a v (v.recOn H0 Hs)
                            def Vector3.append {α : Type u_1} {m : } {n : } (v : Vector3 α m) (w : Vector3 α n) :
                            Vector3 α (n + m)

                            Append two vectors

                            Equations
                            Instances For
                              @[simp]
                              theorem Vector3.append_nil {α : Type u_1} {n : } (w : Vector3 α n) :
                              [].append w = w
                              @[simp]
                              theorem Vector3.append_cons {α : Type u_1} {m : } {n : } (a : α) (v : Vector3 α m) (w : Vector3 α n) :
                              (Vector3.cons a v).append w = Vector3.cons a (v.append w)
                              @[simp]
                              theorem Vector3.append_left {α : Type u_1} {m : } (i : Fin2 m) (v : Vector3 α m) {n : } (w : Vector3 α n) :
                              v.append w (Fin2.left n i) = v i
                              @[simp]
                              theorem Vector3.append_add {α : Type u_1} {m : } (v : Vector3 α m) {n : } (w : Vector3 α n) (i : Fin2 n) :
                              v.append w (i.add m) = w i
                              def Vector3.insert {α : Type u_1} {n : } (a : α) (v : Vector3 α n) (i : Fin2 (n + 1)) :
                              Vector3 α (n + 1)

                              Insert a into v at index i.

                              Equations
                              Instances For
                                @[simp]
                                theorem Vector3.insert_fz {α : Type u_1} {n : } (a : α) (v : Vector3 α n) :
                                Vector3.insert a v Fin2.fz = Vector3.cons a v
                                @[simp]
                                theorem Vector3.insert_fs {α : Type u_1} {n : } (a : α) (b : α) (v : Vector3 α n) (i : Fin2 (n + 1)) :
                                theorem Vector3.append_insert {α : Type u_1} {m : } {n : } (a : α) (t : Vector3 α m) (v : Vector3 α n) (i : Fin2 (n + 1)) (e : n + 1 + m = n + m + 1) :
                                Vector3.insert a (t.append v) (Eq.recOn e (i.add m)) = Eq.recOn e (t.append (Vector3.insert a v i))
                                def VectorEx {α : Type u_1} (k : ) :
                                (Vector3 α kProp)Prop

                                "Curried" exists, i.e. ∃ x₁ ... xₙ, f [x₁, ..., xₙ].

                                Equations
                                Instances For
                                  def VectorAll {α : Type u_1} (k : ) :
                                  (Vector3 α kProp)Prop

                                  "Curried" forall, i.e. ∀ x₁ ... xₙ, f [x₁, ..., xₙ].

                                  Equations
                                  Instances For
                                    theorem exists_vector_zero {α : Type u_1} (f : Vector3 α 0Prop) :
                                    Exists f f []
                                    theorem exists_vector_succ {α : Type u_1} {n : } (f : Vector3 α n.succProp) :
                                    Exists f ∃ (x : α) (v : Vector3 α n), f (Vector3.cons x v)
                                    theorem vectorEx_iff_exists {α : Type u_1} {n : } (f : Vector3 α nProp) :
                                    theorem vectorAll_iff_forall {α : Type u_1} {n : } (f : Vector3 α nProp) :
                                    VectorAll n f ∀ (v : Vector3 α n), f v
                                    def VectorAllP {α : Type u_1} {n : } (p : αProp) (v : Vector3 α n) :

                                    VectorAllP p v is equivalent to ∀ i, p (v i), but unfolds directly to a conjunction, i.e. VectorAllP p [0, 1, 2] = p 0 ∧ p 1 ∧ p 2.

                                    Equations
                                    Instances For
                                      @[simp]
                                      theorem vectorAllP_nil {α : Type u_1} (p : αProp) :
                                      @[simp]
                                      theorem vectorAllP_singleton {α : Type u_1} (p : αProp) (x : α) :
                                      VectorAllP p [x] = p x
                                      @[simp]
                                      theorem vectorAllP_cons {α : Type u_1} {n : } (p : αProp) (x : α) (v : Vector3 α n) :
                                      theorem vectorAllP_iff_forall {α : Type u_1} {n : } (p : αProp) (v : Vector3 α n) :
                                      VectorAllP p v ∀ (i : Fin2 n), p (v i)
                                      theorem VectorAllP.imp {α : Type u_1} {n : } {p : αProp} {q : αProp} (h : ∀ (x : α), p xq x) {v : Vector3 α n} (al : VectorAllP p v) :