Mathlib.LinearAlgebra.VectorSpace

import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.OrderedField.Basic
import Mathlib.Algebra.OrderedField.Square

def vsub.{v} :
  forall (Vector : Sort v), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (x : Vector), forall (y : Vector), Vector :=
  fun Vector => fun vadd => fun vneg => fun x => fun y => vadd x (vneg y)

def linear_comb2.{u,v} :
  forall (Scalar : Sort u), forall (Vector : Sort v), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (a : Scalar), forall (x : Vector), forall (b : Scalar), forall (y : Vector), Vector :=
  fun Scalar => fun Vector => fun vadd => fun smul => fun a => fun x => fun b => fun y => vadd (smul a x) (smul b y)

def linear_comb3.{u,v} :
  forall (Scalar : Sort u), forall (Vector : Sort v), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (a : Scalar), forall (x : Vector), forall (b : Scalar), forall (y : Vector), forall (c : Scalar), forall (z : Vector), Vector :=
  fun Scalar => fun Vector => fun vadd => fun smul => fun a => fun x => fun b => fun y => fun c => fun z => vadd (vadd (smul a x) (smul b y)) (smul c z)

def VectorSpaceLawArgs.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), Prop :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => forall (P : Prop), forall (mk : forall (vec_sub_def_law : forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x y) (vadd x (vneg y))), forall (vec_add_assoc_law : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (vadd (vadd x y) z) (vadd x (vadd y z))), forall (vec_add_comm_law : forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (vadd x y) (vadd y x)), forall (vec_add_zero_law : forall (x : Vector), @Eq.{v} Vector (vadd x vzero) x), forall (vec_zero_add_law : forall (x : Vector), @Eq.{v} Vector (vadd vzero x) x), forall (vec_neg_add_cancel_law : forall (x : Vector), @Eq.{v} Vector (vadd (vneg x) x) vzero), forall (vec_add_neg_cancel_law : forall (x : Vector), @Eq.{v} Vector (vadd x (vneg x)) vzero), forall (sub_sub_sub_cancel_law : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg (@vsub.{v} Vector vadd vneg x z) (@vsub.{v} Vector vadd vneg y z)) (@vsub.{v} Vector vadd vneg x y)), forall (vec_sub_self_law : forall (x : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x x) vzero), forall (vec_sub_zero_law : forall (x : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x vzero) x), forall (vec_add_left_cancel_law : forall (x : Vector), forall (y : Vector), forall (z : Vector), forall (h : @Eq.{v} Vector (vadd x y) (vadd x z)), @Eq.{v} Vector y z), forall (smul_add_law : forall (a : Scalar), forall (b : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (add a b) x) (vadd (smul a x) (smul b x))), forall (add_smul_law : forall (a : Scalar), forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (smul a (vadd x y)) (vadd (smul a x) (smul a y))), forall (one_smul_law : forall (x : Vector), @Eq.{v} Vector (smul one x) x), forall (mul_smul_law : forall (a : Scalar), forall (b : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (mul a b) x) (smul a (smul b x))), forall (zero_smul_law : forall (x : Vector), @Eq.{v} Vector (smul zero x) vzero), forall (smul_zero_law : forall (a : Scalar), @Eq.{v} Vector (smul a vzero) vzero), forall (neg_smul_law : forall (a : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (neg a) x) (vneg (smul a x))), forall (smul_neg_law : forall (a : Scalar), forall (x : Vector), @Eq.{v} Vector (smul a (vneg x)) (vneg (smul a x))), forall (vec_sub_eq_add_neg_law : forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x y) (vadd x (vneg y))), forall (sub_add_sub_cancel_left_law : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (vadd (@vsub.{v} Vector vadd vneg x z) (@vsub.{v} Vector vadd vneg z y)) (@vsub.{v} Vector vadd vneg x y)), forall (linear_comb2_ext_law : forall (a : Scalar), forall (x : Vector), forall (b : Scalar), forall (y : Vector), @Eq.{v} Vector (@linear_comb2.{u,v} Scalar Vector vadd smul a x b y) (vadd (smul a x) (smul b y))), forall (linear_comb3_ext_law : forall (a : Scalar), forall (x : Vector), forall (b : Scalar), forall (y : Vector), forall (c : Scalar), forall (z : Vector), @Eq.{v} Vector (@linear_comb3.{u,v} Scalar Vector vadd smul a x b y c z) (vadd (vadd (smul a x) (smul b y)) (smul c z))), P), P

theorem vec_sub_def.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x y) (vadd x (vneg y)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun x => fun y => @Eq.refl.{v} Vector (@vsub.{v} Vector vadd vneg x y)

theorem vec_add_assoc.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (vadd (vadd x y) z) (vadd x (vadd y z))), forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (vadd (vadd x y) z) (vadd x (vadd y z)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => fun y => fun z => law x y z

theorem vec_add_comm.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (vadd x y) (vadd y x)), forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (vadd x y) (vadd y x) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => fun y => law x y

theorem vec_add_zero.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), @Eq.{v} Vector (vadd x vzero) x), forall (x : Vector), @Eq.{v} Vector (vadd x vzero) x :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => law x

theorem vec_zero_add.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), @Eq.{v} Vector (vadd vzero x) x), forall (x : Vector), @Eq.{v} Vector (vadd vzero x) x :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => law x

theorem vec_neg_add_cancel.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), @Eq.{v} Vector (vadd (vneg x) x) vzero), forall (x : Vector), @Eq.{v} Vector (vadd (vneg x) x) vzero :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => law x

theorem vec_add_neg_cancel.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), @Eq.{v} Vector (vadd x (vneg x)) vzero), forall (x : Vector), @Eq.{v} Vector (vadd x (vneg x)) vzero :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => law x

theorem sub_sub_sub_cancel.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg (@vsub.{v} Vector vadd vneg x z) (@vsub.{v} Vector vadd vneg y z)) (@vsub.{v} Vector vadd vneg x y)), forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg (@vsub.{v} Vector vadd vneg x z) (@vsub.{v} Vector vadd vneg y z)) (@vsub.{v} Vector vadd vneg x y) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => fun y => fun z => law x y z

theorem vec_sub_self.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x x) vzero), forall (x : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x x) vzero :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => law x

theorem vec_sub_zero.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x vzero) x), forall (x : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x vzero) x :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => law x

theorem vec_add_left_cancel.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), forall (y : Vector), forall (z : Vector), forall (h : @Eq.{v} Vector (vadd x y) (vadd x z)), @Eq.{v} Vector y z), forall (x : Vector), forall (y : Vector), forall (z : Vector), forall (h : @Eq.{v} Vector (vadd x y) (vadd x z)), @Eq.{v} Vector y z :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => fun y => fun z => fun h => law x y z h

theorem smul_add.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (a : Scalar), forall (b : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (add a b) x) (vadd (smul a x) (smul b x))), forall (a : Scalar), forall (b : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (add a b) x) (vadd (smul a x) (smul b x)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun a => fun b => fun x => law a b x

theorem add_smul.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (a : Scalar), forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (smul a (vadd x y)) (vadd (smul a x) (smul a y))), forall (a : Scalar), forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (smul a (vadd x y)) (vadd (smul a x) (smul a y)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun a => fun x => fun y => law a x y

theorem one_smul.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), @Eq.{v} Vector (smul one x) x), forall (x : Vector), @Eq.{v} Vector (smul one x) x :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => law x

theorem mul_smul.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (a : Scalar), forall (b : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (mul a b) x) (smul a (smul b x))), forall (a : Scalar), forall (b : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (mul a b) x) (smul a (smul b x)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun a => fun b => fun x => law a b x

theorem zero_smul.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), @Eq.{v} Vector (smul zero x) vzero), forall (x : Vector), @Eq.{v} Vector (smul zero x) vzero :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => law x

theorem smul_zero.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (a : Scalar), @Eq.{v} Vector (smul a vzero) vzero), forall (a : Scalar), @Eq.{v} Vector (smul a vzero) vzero :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun a => law a

theorem neg_smul.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (a : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (neg a) x) (vneg (smul a x))), forall (a : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (neg a) x) (vneg (smul a x)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun a => fun x => law a x

theorem smul_neg.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (a : Scalar), forall (x : Vector), @Eq.{v} Vector (smul a (vneg x)) (vneg (smul a x))), forall (a : Scalar), forall (x : Vector), @Eq.{v} Vector (smul a (vneg x)) (vneg (smul a x)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun a => fun x => law a x

theorem vec_sub_eq_add_neg.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x y) (vadd x (vneg y)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun x => fun y => @Eq.refl.{v} Vector (@vsub.{v} Vector vadd vneg x y)

theorem sub_add_sub_cancel_left.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (law : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (vadd (@vsub.{v} Vector vadd vneg x z) (@vsub.{v} Vector vadd vneg z y)) (@vsub.{v} Vector vadd vneg x y)), forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (vadd (@vsub.{v} Vector vadd vneg x z) (@vsub.{v} Vector vadd vneg z y)) (@vsub.{v} Vector vadd vneg x y) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun law => fun x => fun y => fun z => law x y z

theorem linear_comb2_ext.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (a : Scalar), forall (x : Vector), forall (b : Scalar), forall (y : Vector), @Eq.{v} Vector (@linear_comb2.{u,v} Scalar Vector vadd smul a x b y) (vadd (smul a x) (smul b y)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun a => fun x => fun b => fun y => @Eq.refl.{v} Vector (@linear_comb2.{u,v} Scalar Vector vadd smul a x b y)

theorem linear_comb3_ext.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (a : Scalar), forall (x : Vector), forall (b : Scalar), forall (y : Vector), forall (c : Scalar), forall (z : Vector), @Eq.{v} Vector (@linear_comb3.{u,v} Scalar Vector vadd smul a x b y c z) (vadd (vadd (smul a x) (smul b y)) (smul c z)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun a => fun x => fun b => fun y => fun c => fun z => @Eq.refl.{v} Vector (@linear_comb3.{u,v} Scalar Vector vadd smul a x b y c z)