Mathlib.Geometry.Metric.Abstract

import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.OrderedField.Basic
import Mathlib.Algebra.OrderedField.Square
import Mathlib.LinearAlgebra.VectorSpace
import Mathlib.LinearAlgebra.InnerProduct
import Mathlib.LinearAlgebra.InnerProduct.Derived
import Mathlib.Geometry.Affine
import Mathlib.Geometry.Affine.Derived
import Mathlib.Geometry.RightTriangle.Abstract

def dist.{p,u,v} :
  forall (Scalar : Sort u), forall (sqrt_fn : forall (a : Scalar), Scalar), 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 sqrt_fn => fun Vector => fun inner => fun PointCarrier => fun disp_op => fun A => fun B => @sqrt.{u} Scalar sqrt_fn (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B)

def MetricSpaceLawArgs.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (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 lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => forall (P : Prop), forall (mk : forall (dist_def_law : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@sqrt.{u} Scalar sqrt_fn (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B))), forall (dist_nonneg_law : forall (A : PointCarrier), forall (B : PointCarrier), le_rel zero (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)), forall (distance_symm_law : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B A)), forall (distance_zero_iff_eq_law : forall (A : PointCarrier), forall (B : PointCarrier), forall (R : Prop), forall (mk : forall (forward : forall (h : @Eq.{u} Scalar (@dist.{p,u,v} Scalar sqrt_fn 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 (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) zero), R), R), forall (triangle_inequality_law : forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), le_rel (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C) (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C))), P), P

def Ball.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (center : PointCarrier), forall (radius : Scalar), forall (x : PointCarrier), Prop :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun center => fun radius => fun x => le_rel (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op center x) radius

theorem dist_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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (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 (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@sqrt.{u} Scalar sqrt_fn (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => 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 (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)

theorem point_dist_sq_nonneg_from_inner_args.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (A : PointCarrier), forall (B : PointCarrier), le_rel zero (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun inner_args => fun A => fun B => inner_args (le_rel zero (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B)) (fun (dot_comm_arg : forall (x : Vector), forall (y : Vector), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) (@dot.{u,v} Scalar Vector inner y x)) => fun (dot_add_left_arg : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner (vadd x y) z) (add (@dot.{u,v} Scalar Vector inner x z) (@dot.{u,v} Scalar Vector inner y z))) => fun (dot_add_right_arg : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x (vadd y z)) (add (@dot.{u,v} Scalar Vector inner x y) (@dot.{u,v} Scalar Vector inner x z))) => fun (dot_neg_left_arg : forall (x : Vector), forall (y : Vector), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner (vneg x) y) (neg (@dot.{u,v} Scalar Vector inner x y))) => fun (dot_neg_right_arg : forall (x : Vector), forall (y : Vector), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x (vneg y)) (neg (@dot.{u,v} Scalar Vector inner x y))) => fun (dot_sub_left_arg : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner (@vsub.{v} Vector vadd vneg x y) z) (sub (@dot.{u,v} Scalar Vector inner x z) (@dot.{u,v} Scalar Vector inner y z))) => fun (dot_sub_right_arg : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x (@vsub.{v} Vector vadd vneg y z)) (sub (@dot.{u,v} Scalar Vector inner x y) (@dot.{u,v} Scalar Vector inner x z))) => fun (norm_sq_def_arg : forall (x : Vector), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner x) (@dot.{u,v} Scalar Vector inner x x)) => fun (dist_sq_def_arg : forall (x : Vector), forall (y : Vector), @Eq.{u} Scalar (@distSq.{u,v} Scalar Vector vadd vneg inner x y) (@normSq.{u,v} Scalar Vector inner (@vsub.{v} Vector vadd vneg y x))) => fun (dot_self_eq_norm_sq_arg : forall (x : Vector), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x x) (@normSq.{u,v} Scalar Vector inner x)) => fun (norm_sq_add_arg : forall (x : Vector), forall (y : Vector), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vadd x y)) (add (add (@normSq.{u,v} Scalar Vector inner x) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner x y))) (@normSq.{u,v} Scalar Vector inner y))) => fun (norm_sq_sub_arg : forall (x : Vector), forall (y : Vector), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (@vsub.{v} Vector vadd vneg x y)) (add (sub (@normSq.{u,v} Scalar Vector inner x) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner x y))) (@normSq.{u,v} Scalar Vector inner y))) => fun (inner_field13_arg : forall (x : Vector), forall (y : Vector), forall (h : @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vadd x y)) (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) => fun (inner_field14_arg : forall (x : Vector), forall (y : Vector), forall (h : @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (@vsub.{v} Vector vadd vneg x y)) (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) => fun (norm_sq_nonneg_arg : forall (x : Vector), le_rel zero (@normSq.{u,v} Scalar Vector inner x)) => fun (parallelogram_arg : forall (x : Vector), forall (y : Vector), @Eq.{u} Scalar (add (@normSq.{u,v} Scalar Vector inner (vadd x y)) (@normSq.{u,v} Scalar Vector inner (@vsub.{v} Vector vadd vneg x y))) (add (mul (@two.{u} Scalar one add) (@normSq.{u,v} Scalar Vector inner x)) (mul (@two.{u} Scalar one add) (@normSq.{u,v} Scalar Vector inner y)))) => fun (polarization_identity_arg : forall (x : Vector), forall (y : Vector), @Eq.{u} Scalar (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner x y)) (sub (@normSq.{u,v} Scalar Vector inner (vadd x y)) (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)))) => fun (perp_vec_iff_dot_eq_zero_arg : forall (x : Vector), forall (y : Vector), forall (R : Prop), forall (mk : forall (forward : forall (h : @PerpVec.{u,v} Scalar zero Vector inner x y), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero), forall (backward : forall (h : @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero), @PerpVec.{u,v} Scalar zero Vector inner x y), R), R) => fun (perp_vec_symm_arg : forall (x : Vector), forall (y : Vector), forall (h : @PerpVec.{u,v} Scalar zero Vector inner x y), @PerpVec.{u,v} Scalar zero Vector inner y x) => fun (norm_sq_zero_iff_arg : forall (x : Vector), forall (R : Prop), forall (mk : forall (forward : forall (h : @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner x) zero), @Eq.{v} Vector x vzero), forall (backward : forall (h : @Eq.{v} Vector x vzero), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner x) zero), R), R) => fun (dist_sq_nonneg_arg : forall (x : Vector), forall (y : Vector), le_rel zero (@distSq.{u,v} Scalar Vector vadd vneg inner x y)) => fun (inner_field23_arg : forall (x : Vector), forall (y : Vector), forall (h : @PerpVec.{u,v} Scalar zero Vector inner x y), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vadd x y)) (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) => fun (inner_field24_arg : forall (x : Vector), forall (y : Vector), forall (h : @PerpVec.{u,v} Scalar zero Vector inner x y), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (@vsub.{v} Vector vadd vneg x y)) (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) => fun (quadratic_norm_nonneg_arg : forall (x : Vector), forall (y : Vector), forall (t : Scalar), le_rel zero (add (add (mul (@normSq.{u,v} Scalar Vector inner x) (@sq.{u} Scalar mul t)) (mul (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner x y)) t)) (@normSq.{u,v} Scalar Vector inner y))) => norm_sq_nonneg_arg (@disp.{p,v} PointCarrier Vector disp_op A B))

theorem square_dist_eq_dist_sq_from_law_packages.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ordered_args : @OrderedFieldLawArgs.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ordered_args => fun inner_args => fun A => fun B => ordered_args (@Eq.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B)) (fun (le_refl_arg : forall (a : Scalar), le_rel a a) => fun (le_trans_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (hab : le_rel a b), forall (hbc : le_rel b c), le_rel a c) => fun (add_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), le_rel zero (add a b)) => fun (mul_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), le_rel zero (mul a b)) => fun (square_nonneg_arg : forall (a : Scalar), le_rel zero (@sq.{u} Scalar mul a)) => fun (sqrt_nonneg_arg : forall (a : Scalar), le_rel zero (sqrt_fn a)) => fun (sqrt_square_of_nonneg_arg : forall (a : Scalar), forall (ha : le_rel zero a), @Eq.{u} Scalar (sqrt_fn (@sq.{u} Scalar mul a)) a) => fun (sqrt_mul_self_arg : forall (a : Scalar), forall (ha : le_rel zero a), @Eq.{u} Scalar (sqrt_fn (mul a a)) a) => fun (eq_of_square_eq_square_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), forall (hsq : @Eq.{u} Scalar (@sq.{u} Scalar mul a) (@sq.{u} Scalar mul b)), @Eq.{u} Scalar a b) => fun (add_le_add_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (d : Scalar), forall (hab : le_rel a b), forall (hcd : le_rel c d), le_rel (add a c) (add b d)) => fun (mul_le_mul_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (d : Scalar), forall (ha : le_rel zero a), forall (hab : le_rel a b), forall (hc : le_rel zero c), forall (hcd : le_rel c d), le_rel (mul a c) (mul b d)) => fun (zero_le_two_arg : le_rel zero (@two.{u} Scalar one add)) => fun (le_antisymm_arg : forall (a : Scalar), forall (b : Scalar), forall (hab : le_rel a b), forall (hba : le_rel b a), @Eq.{u} Scalar a b) => fun (lt_of_le_of_ne_arg : forall (a : Scalar), forall (ha : le_rel zero a), forall (hne : forall (haz : @Eq.{u} Scalar a zero), forall (P : Prop), P), lt_rel zero a) => fun (le_of_eq_arg : forall (a : Scalar), forall (b : Scalar), forall (hab : @Eq.{u} Scalar a b), forall (P : Prop), forall (mk : forall (hab_le : le_rel a b), forall (hba_le : le_rel b a), P), P) => fun (sqrt_sq_arg : forall (a : Scalar), forall (ha : le_rel zero a), @Eq.{u} Scalar (@sq.{u} Scalar mul (sqrt_fn a)) a) => fun (sq_eq_zero_iff_arg : forall (a : Scalar), forall (R : Prop), forall (mk : forall (forward : forall (hsqz : @Eq.{u} Scalar (@sq.{u} Scalar mul a) zero), @Eq.{u} Scalar a zero), forall (backward : forall (haz : @Eq.{u} Scalar a zero), @Eq.{u} Scalar (@sq.{u} Scalar mul a) zero), R), R) => fun (sum_nonneg_eq_zero_arg : forall (a : Scalar), forall (b : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), forall (hsum : @Eq.{u} Scalar (add a b) zero), forall (R : Prop), forall (mk : forall (haz : @Eq.{u} Scalar a zero), forall (hbz : @Eq.{u} Scalar b zero), R), R) => fun (square_completion_bound_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (hquadratic : forall (t : Scalar), le_rel zero (add (add (mul a (@sq.{u} Scalar mul t)) (mul (mul (@two.{u} Scalar one add) b) t)) c)), le_rel (@sq.{u} Scalar mul b) (mul a c)) => fun (le_of_sq_le_sq_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), forall (hsq : le_rel (@sq.{u} Scalar mul a) (@sq.{u} Scalar mul b)), le_rel a b) => fun (le_mul_sqrt_of_sq_le_mul_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), forall (hsq : le_rel (@sq.{u} Scalar mul c) (mul a b)), le_rel c (mul (sqrt_fn a) (sqrt_fn b))) => fun (add_two_mul_le_sq_add_sqrt_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (n : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), forall (hn : @Eq.{u} Scalar n (add (add a (mul (@two.{u} Scalar one add) c)) b)), forall (hc : le_rel c (mul (sqrt_fn a) (sqrt_fn b))), le_rel n (@sq.{u} Scalar mul (add (sqrt_fn a) (sqrt_fn b)))) => sqrt_sq_arg (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@point_dist_sq_nonneg_from_inner_args.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op inner_args A B))

theorem dist_sq_eq_square_dist_from_law_packages.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ordered_args : @OrderedFieldLawArgs.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ordered_args => fun inner_args => fun A => fun B => @Eq.rec.{u,0} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (fun (q : Scalar) => fun (hq : @Eq.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) q) => @Eq.{u} Scalar q (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B))) (@Eq.refl.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B))) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@square_dist_eq_dist_sq_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args inner_args A B)

theorem dist_sq_eq_square_dist.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ordered_args : @OrderedFieldLawArgs.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ordered_args => fun inner_args => fun A => fun B => @dist_sq_eq_square_dist_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args inner_args A B

theorem dist_sq_points_le_square_sum_dist_from_law_packages.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (ordered_args : @OrderedFieldLawArgs.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), le_rel (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@sq.{u} Scalar mul (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C))) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ring_args => fun ordered_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => @Eq.rec.{u,0} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (fun (q : Scalar) => fun (hq : @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) q) => forall (hbound : le_rel q (@sq.{u} Scalar mul (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)))), le_rel (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@sq.{u} Scalar mul (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)))) (fun (hbound : le_rel (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@sq.{u} Scalar mul (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)))) => hbound) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A C)) (@dist_sq_points_def_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A C) (@Eq.rec.{v,0} Vector (@disp.{p,v} PointCarrier Vector disp_op A C) (fun (z : Vector) => fun (hz : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A C) z) => forall (hbound : le_rel (@normSq.{u,v} Scalar Vector inner z) (@sq.{u} Scalar mul (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)))), le_rel (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A C)) (@sq.{u} Scalar mul (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)))) (fun (hbound : le_rel (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A C)) (@sq.{u} Scalar mul (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)))) => hbound) (vadd (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op B C)) (@disp_comp_from_affine_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A B C) (@norm_sq_add_le_square_sum_norms_from_cauchy.{u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner ring_args ordered_args vector_args inner_args (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op B C)))

theorem dist_nonneg_from_ordered_args.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ordered_args : @OrderedFieldLawArgs.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn), forall (A : PointCarrier), forall (B : PointCarrier), le_rel zero (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ordered_args => fun A => fun B => ordered_args (le_rel zero (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (fun (le_refl_arg : forall (a : Scalar), le_rel a a) => fun (le_trans_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (hab : le_rel a b), forall (hbc : le_rel b c), le_rel a c) => fun (add_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), le_rel zero (add a b)) => fun (mul_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), le_rel zero (mul a b)) => fun (square_nonneg_arg : forall (a : Scalar), le_rel zero (@sq.{u} Scalar mul a)) => fun (sqrt_nonneg_arg : forall (a : Scalar), le_rel zero (sqrt_fn a)) => fun (sqrt_square_of_nonneg_arg : forall (a : Scalar), forall (ha : le_rel zero a), @Eq.{u} Scalar (sqrt_fn (@sq.{u} Scalar mul a)) a) => fun (sqrt_mul_self_arg : forall (a : Scalar), forall (ha : le_rel zero a), @Eq.{u} Scalar (sqrt_fn (mul a a)) a) => fun (eq_of_square_eq_square_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), forall (hsq : @Eq.{u} Scalar (@sq.{u} Scalar mul a) (@sq.{u} Scalar mul b)), @Eq.{u} Scalar a b) => fun (add_le_add_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (d : Scalar), forall (hab : le_rel a b), forall (hcd : le_rel c d), le_rel (add a c) (add b d)) => fun (mul_le_mul_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (d : Scalar), forall (ha : le_rel zero a), forall (hab : le_rel a b), forall (hc : le_rel zero c), forall (hcd : le_rel c d), le_rel (mul a c) (mul b d)) => fun (zero_le_two_arg : le_rel zero (@two.{u} Scalar one add)) => fun (le_antisymm_arg : forall (a : Scalar), forall (b : Scalar), forall (hab : le_rel a b), forall (hba : le_rel b a), @Eq.{u} Scalar a b) => fun (lt_of_le_of_ne_arg : forall (a : Scalar), forall (ha : le_rel zero a), forall (hne : forall (haz : @Eq.{u} Scalar a zero), forall (P : Prop), P), lt_rel zero a) => fun (le_of_eq_arg : forall (a : Scalar), forall (b : Scalar), forall (hab : @Eq.{u} Scalar a b), forall (P : Prop), forall (mk : forall (hab_le : le_rel a b), forall (hba_le : le_rel b a), P), P) => fun (sqrt_sq_arg : forall (a : Scalar), forall (ha : le_rel zero a), @Eq.{u} Scalar (@sq.{u} Scalar mul (sqrt_fn a)) a) => fun (sq_eq_zero_iff_arg : forall (a : Scalar), forall (R : Prop), forall (mk : forall (forward : forall (hsqz : @Eq.{u} Scalar (@sq.{u} Scalar mul a) zero), @Eq.{u} Scalar a zero), forall (backward : forall (haz : @Eq.{u} Scalar a zero), @Eq.{u} Scalar (@sq.{u} Scalar mul a) zero), R), R) => fun (sum_nonneg_eq_zero_arg : forall (a : Scalar), forall (b : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), forall (hsum : @Eq.{u} Scalar (add a b) zero), forall (R : Prop), forall (mk : forall (haz : @Eq.{u} Scalar a zero), forall (hbz : @Eq.{u} Scalar b zero), R), R) => fun (square_completion_bound_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (hquadratic : forall (t : Scalar), le_rel zero (add (add (mul a (@sq.{u} Scalar mul t)) (mul (mul (@two.{u} Scalar one add) b) t)) c)), le_rel (@sq.{u} Scalar mul b) (mul a c)) => fun (le_of_sq_le_sq_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), forall (hsq : le_rel (@sq.{u} Scalar mul a) (@sq.{u} Scalar mul b)), le_rel a b) => fun (le_mul_sqrt_of_sq_le_mul_nonneg_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), forall (hsq : le_rel (@sq.{u} Scalar mul c) (mul a b)), le_rel c (mul (sqrt_fn a) (sqrt_fn b))) => fun (add_two_mul_le_sq_add_sqrt_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (n : Scalar), forall (ha : le_rel zero a), forall (hb : le_rel zero b), forall (hn : @Eq.{u} Scalar n (add (add a (mul (@two.{u} Scalar one add) c)) b)), forall (hc : le_rel c (mul (sqrt_fn a) (sqrt_fn b))), le_rel n (@sq.{u} Scalar mul (add (sqrt_fn a) (sqrt_fn b)))) => sqrt_nonneg_arg (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B))

theorem triangle_inequality_from_law_packages.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (ordered_args : @OrderedFieldLawArgs.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), le_rel (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C) (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ring_args => fun ordered_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => @sqrt_sum_square_bound_from_ordered_args.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn ordered_args (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C) (@dist_nonneg_from_ordered_args.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args A C) (@dist_nonneg_from_ordered_args.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args A B) (@dist_nonneg_from_ordered_args.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args B C) (@Eq.rec.{u,0} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C)) (fun (q : Scalar) => fun (hq : @Eq.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C)) q) => forall (hbound : le_rel q (@sq.{u} Scalar mul (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)))), le_rel (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C)) (@sq.{u} Scalar mul (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)))) (fun (hbound : le_rel (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C)) (@sq.{u} Scalar mul (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)))) => hbound) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@square_dist_eq_dist_sq_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args inner_args A C) (@dist_sq_points_le_square_sum_dist_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ring_args ordered_args vector_args inner_args affine_args A B C))

theorem dist_nonneg.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (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), le_rel zero (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)), forall (A : PointCarrier), forall (B : PointCarrier), le_rel zero (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => 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 distance_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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (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 (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B A)), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn 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 lt_rel => fun sqrt_fn => 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 distance_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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (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 (@dist.{p,u,v} Scalar sqrt_fn 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 (@dist.{p,u,v} Scalar sqrt_fn 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 (@dist.{p,u,v} Scalar sqrt_fn 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 (@dist.{p,u,v} Scalar sqrt_fn 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 lt_rel => fun sqrt_fn => 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 (@dist.{p,u,v} Scalar sqrt_fn 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 (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) zero), R) => law A B R mk

theorem pythagorean_distance_general.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (metric_pythagorean_target : forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), forall (h : @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C), @Eq.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)) (add (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C)))), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), forall (h : @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C), @Eq.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)) (add (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C))) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun metric_pythagorean_target => fun A => fun B => fun C => fun h => metric_pythagorean_target A B C h

theorem triangle_inequality.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : 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 (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), le_rel (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C) (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C))), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), le_rel (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C) (add (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B) (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => 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