Mathlib.Geometry.Affine

import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.OrderedField.Basic
import Mathlib.Algebra.OrderedField.Square
import Mathlib.LinearAlgebra.VectorSpace
import Mathlib.LinearAlgebra.InnerProduct

def Point.{p} :
  forall (Carrier : Sort p), Sort p :=
  fun Carrier => Carrier

def disp.{p,v} :
  forall (PointCarrier : Sort p), forall (Vector : Sort v), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (A : PointCarrier), forall (B : PointCarrier), Vector :=
  fun PointCarrier => fun Vector => fun disp_op => fun A => fun B => disp_op A B

def distSqPoints.{p,u,v} :
  forall (Scalar : Sort u), forall (Vector : Sort v), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (A : PointCarrier), forall (B : PointCarrier), Scalar :=
  fun Scalar => fun Vector => fun inner => fun PointCarrier => fun disp_op => fun A => fun B => @normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B)

def translate.{p,v} :
  forall (PointCarrier : Sort p), forall (Vector : Sort v), forall (translate_op : forall (A : PointCarrier), forall (v : Vector), PointCarrier), forall (A : PointCarrier), forall (v : Vector), PointCarrier :=
  fun PointCarrier => fun Vector => fun translate_op => fun A => fun v => translate_op A v

def midpoint.{p} :
  forall (PointCarrier : Sort p), forall (midpoint_op : forall (A : PointCarrier), forall (B : PointCarrier), PointCarrier), forall (A : PointCarrier), forall (B : PointCarrier), PointCarrier :=
  fun PointCarrier => fun midpoint_op => fun A => fun B => midpoint_op A B

def collinear.{p} :
  forall (PointCarrier : Sort p), forall (collinear_rel : forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), Prop), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), Prop :=
  fun PointCarrier => fun collinear_rel => fun A => fun B => fun C => collinear_rel A B C

def AffineLawArgs.{p,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 (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), Prop :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => forall (P : Prop), forall (mk : forall (disp_self_law : forall (A : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A A) vzero), forall (disp_reverse_law : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B A) (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))), forall (disp_comp_law : forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A C) (vadd (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op B C))), forall (point_ext_of_zero_disp_law : forall (A : PointCarrier), forall (B : PointCarrier), forall (h : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A B) vzero), @Eq.{p} PointCarrier A B), forall (dist_sq_symm_law : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A)), forall (dist_sq_zero_iff_eq_law : forall (A : PointCarrier), forall (B : PointCarrier), forall (R : Prop), forall (mk : forall (forward : forall (h : @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), @Eq.{p} PointCarrier A B), forall (backward : forall (h : @Eq.{p} PointCarrier A B), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), R), R), P), P

theorem disp_self.{p,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 (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (law : forall (A : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A A) vzero), forall (A : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A A) vzero :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun law => fun A => law A

theorem disp_reverse.{p,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 (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (law : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B A) (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B A) (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun law => fun A => fun B => law A B

theorem disp_comp.{p,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 (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (law : forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A C) (vadd (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op B C))), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A C) (vadd (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op B C)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun law => fun A => fun B => fun C => law A B C

theorem hypotenuse_vector_eq_sub_legs.{p,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 (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (law : forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) (@vsub.{v} Vector vadd vneg (@disp.{p,v} PointCarrier Vector disp_op A C) (@disp.{p,v} PointCarrier Vector disp_op A B))), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) (@vsub.{v} Vector vadd vneg (@disp.{p,v} PointCarrier Vector disp_op A C) (@disp.{p,v} PointCarrier Vector disp_op A B)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun law => fun A => fun B => fun C => law A B C

theorem dist_sq_points_def.{p,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 (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun A => fun B => @Eq.refl.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B)

theorem point_ext_of_zero_disp.{p,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 (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (law : forall (A : PointCarrier), forall (B : PointCarrier), forall (h : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A B) vzero), @Eq.{p} PointCarrier A B), forall (A : PointCarrier), forall (B : PointCarrier), forall (h : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A B) vzero), @Eq.{p} PointCarrier A B :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun law => fun A => fun B => fun h => law A B h

theorem dist_sq_symm.{p,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 (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (law : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A)), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun law => fun A => fun B => law A B

theorem dist_sq_zero_iff_eq.{p,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 (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (law : forall (A : PointCarrier), forall (B : PointCarrier), forall (R : Prop), forall (mk : forall (forward : forall (h : @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), @Eq.{p} PointCarrier A B), forall (backward : forall (h : @Eq.{p} PointCarrier A B), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), R), R), forall (A : PointCarrier), forall (B : PointCarrier), forall (R : Prop), forall (mk : forall (forward : forall (h : @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), @Eq.{p} PointCarrier A B), forall (backward : forall (h : @Eq.{p} PointCarrier A B), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), R), R :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun law => fun A => fun B => fun (R : Prop) => fun (mk : forall (forward : forall (h : @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), @Eq.{p} PointCarrier A B), forall (backward : forall (h : @Eq.{p} PointCarrier A B), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), R) => law A B R mk