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.

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

    The empty vector

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        @[match_pattern]
        def Vector3.cons {α : Type u_1} {n : } (a : α) (v : Vector3 α n) :
        Vector3 α (n + 1)

        The vector cons operation

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          • One or more equations did not get rendered due to their size.
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            • One or more equations did not get rendered due to their size.
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              The vector cons operation

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                @[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

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

                  Construct a vector from a function on Fin2.

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                    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
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                      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.

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                          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.

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                            @[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.

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                              @[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

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                                @[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.

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                                  @[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].

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

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

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                                      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.

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                                        @[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) :