Mathlib.LinearAlgebra.InnerProduct.Derived

import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.OrderedField.Basic
import Mathlib.Algebra.OrderedField.ScalarIdentities
import Mathlib.Logic.EqReasoning
import Mathlib.LinearAlgebra.VectorSpace
import Mathlib.LinearAlgebra.InnerProduct

theorem norm_sq_add_from_inner_args.{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 (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), 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 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 inner_args => fun x => fun y => inner_args (@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 (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_add_arg x y)

theorem norm_sq_sub_from_inner_args.{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 (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), 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 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 inner_args => fun x => fun y => inner_args (@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 (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_sub_arg x y)

theorem parallelogram_law_from_inner_args.{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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), 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 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 ring_args => fun inner_args => fun x => fun y => @eq_trans.{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 (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)) (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))) (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))) (@eq_congr2.{u,u,u} Scalar Scalar Scalar add (@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)) (@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)) (@norm_sq_add_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args x y) (@norm_sq_sub_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args x y)) (@parallelogram_scalar_rhs_from_ring_args.{u} Scalar zero one add neg sub mul ring_args (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner x y)))

theorem polarization_identity_from_inner_args.{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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), 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 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 ring_args => fun inner_args => fun x => fun y => @eq_trans.{u} Scalar (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner x y)) (sub (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)) (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner 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))) (@polarization_scalar_rhs_from_ring_args.{u} Scalar zero one add neg sub mul ring_args (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y) (@dot.{u,v} Scalar Vector inner x y)) (@eq_congr2.{u,u,u} Scalar Scalar Scalar sub (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)) (@normSq.{u,v} Scalar Vector inner (vadd x y)) (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (@eq_symm.{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)) (@norm_sq_add_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args x y)) (@Eq.refl.{u} Scalar (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))))

theorem dot_neg_left_from_inner_args.{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 (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), 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 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 inner_args => fun x => fun y => inner_args (@Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner (vneg x) y) (neg (@dot.{u,v} Scalar Vector inner x y))) (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))) => dot_neg_left_arg x y)

theorem norm_sq_neg_from_inner_args.{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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (x : Vector), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vneg x)) (@normSq.{u,v} Scalar Vector inner x) :=
  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 ring_args => fun inner_args => fun x => ring_args (@Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vneg x)) (@normSq.{u,v} Scalar Vector inner x)) (fun (sub_eq_add_neg_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (sub a b) (add a (neg b))) => fun (add_assoc_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (add (add a b) c) (add a (add b c))) => fun (add_comm_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (add a b) (add b a)) => fun (add_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (add a zero) a) => fun (zero_add_arg : forall (a : Scalar), @Eq.{u} Scalar (add zero a) a) => fun (neg_add_cancel_arg : forall (a : Scalar), @Eq.{u} Scalar (add (neg a) a) zero) => fun (add_neg_cancel_arg : forall (a : Scalar), @Eq.{u} Scalar (add a (neg a)) zero) => fun (sub_self_arg : forall (a : Scalar), @Eq.{u} Scalar (sub a a) zero) => fun (mul_assoc_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul (mul a b) c) (mul a (mul b c))) => fun (mul_comm_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (mul a b) (mul b a)) => fun (mul_one_arg : forall (a : Scalar), @Eq.{u} Scalar (mul a one) a) => fun (one_mul_arg : forall (a : Scalar), @Eq.{u} Scalar (mul one a) a) => fun (left_distrib_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul a (add b c)) (add (mul a b) (mul a c))) => fun (right_distrib_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul (add a b) c) (add (mul a c) (mul b c))) => fun (mul_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (mul a zero) zero) => fun (zero_mul_arg : forall (a : Scalar), @Eq.{u} Scalar (mul zero a) zero) => fun (add_left_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (h : @Eq.{u} Scalar (add a b) (add a c)), @Eq.{u} Scalar b c) => fun (ring_normalize_add_mul3_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (add (add (mul a b) (mul b c)) (mul a c)) (add (add (mul a b) (mul a c)) (mul b c))) => fun (add_right_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (h : @Eq.{u} Scalar (add b a) (add c a)), @Eq.{u} Scalar b c) => fun (neg_neg_arg : forall (a : Scalar), @Eq.{u} Scalar (neg (neg a)) a) => fun (sub_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (sub a zero) a) => fun (zero_sub_arg : forall (a : Scalar), @Eq.{u} Scalar (sub zero a) (neg a)) => fun (sub_add_cancel_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (add (sub a b) b) a) => fun (add_sub_cancel_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (sub (add a b) b) a) => fun (sub_add_sub_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (sub (sub a c) (sub b c)) (sub a b)) => inner_args (@Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vneg x)) (@normSq.{u,v} Scalar Vector inner x)) (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))) => @Eq.rec.{u,0} Scalar (@dot.{u,v} Scalar Vector inner (vneg x) (vneg x)) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner (vneg x) (vneg x)) z) => @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vneg x)) z) (norm_sq_def_arg (vneg x)) (@normSq.{u,v} Scalar Vector inner x) (@Eq.rec.{u,0} Scalar (neg (@dot.{u,v} Scalar Vector inner x (vneg x))) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (neg (@dot.{u,v} Scalar Vector inner x (vneg x))) z) => @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner (vneg x) (vneg x)) z) (@dot_neg_left_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args x (vneg x)) (@normSq.{u,v} Scalar Vector inner x) (@Eq.rec.{u,0} Scalar (neg (neg (@dot.{u,v} Scalar Vector inner x x))) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (neg (neg (@dot.{u,v} Scalar Vector inner x x))) z) => @Eq.{u} Scalar (neg (@dot.{u,v} Scalar Vector inner x (vneg x))) z) (@Eq.rec.{u,0} Scalar (@dot.{u,v} Scalar Vector inner x (vneg x)) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x (vneg x)) z) => @Eq.{u} Scalar (neg (@dot.{u,v} Scalar Vector inner x (vneg x))) (neg z)) (@Eq.refl.{u} Scalar (neg (@dot.{u,v} Scalar Vector inner x (vneg x)))) (neg (@dot.{u,v} Scalar Vector inner x x)) (dot_neg_right_arg x x)) (@normSq.{u,v} Scalar Vector inner x) (@Eq.rec.{u,0} Scalar (@dot.{u,v} Scalar Vector inner x x) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x x) z) => @Eq.{u} Scalar (neg (neg (@dot.{u,v} Scalar Vector inner x x))) z) (neg_neg_arg (@dot.{u,v} Scalar Vector inner x x)) (@normSq.{u,v} Scalar Vector inner x) (dot_self_eq_norm_sq_arg x))))))

theorem norm_sq_add_of_dot_zero_from_args.{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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (x : Vector), forall (y : Vector), forall (hzero : @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 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 ring_args => fun inner_args => fun x => fun y => fun hzero => @Eq.rec.{u,0} Scalar (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 (z : Scalar) => fun (hz : @Eq.{u} Scalar (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)) z) => @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vadd x y)) z) (@norm_sq_add_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args x y) (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (@normalize_add_with_zero_cross_term_from_ring_args.{u} Scalar zero one add neg sub mul ring_args (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y) (@dot.{u,v} Scalar Vector inner x y) hzero)

theorem norm_sq_add_of_perp_from_args.{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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (x : Vector), forall (y : Vector), forall (hperp : @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 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 ring_args => fun inner_args => fun x => fun y => fun hperp => @norm_sq_add_of_dot_zero_from_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner ring_args inner_args x y hperp

theorem norm_sq_add_neg_left_from_inner_args.{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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (x : Vector), forall (y : Vector), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vadd (vneg x) y)) (sub (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner x y))) :=
  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 ring_args => fun inner_args => fun x => fun y => @Eq.rec.{u,0} Scalar (add (add (@normSq.{u,v} Scalar Vector inner (vneg x)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y)) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (add (add (@normSq.{u,v} Scalar Vector inner (vneg x)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y)) z) => @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vadd (vneg x) y)) z) (@norm_sq_add_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args (vneg x) y) (sub (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner x y))) (@Eq.rec.{u,0} Scalar (add (add (@normSq.{u,v} Scalar Vector inner x) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y)) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (add (add (@normSq.{u,v} Scalar Vector inner x) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y)) z) => @Eq.{u} Scalar (add (add (@normSq.{u,v} Scalar Vector inner (vneg x)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y)) z) (@Eq.rec.{u,0} Scalar (@normSq.{u,v} Scalar Vector inner (vneg x)) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vneg x)) z) => @Eq.{u} Scalar (add (add (@normSq.{u,v} Scalar Vector inner (vneg x)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y)) (add (add z (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y))) (@Eq.refl.{u} Scalar (add (add (@normSq.{u,v} Scalar Vector inner (vneg x)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y))) (@normSq.{u,v} Scalar Vector inner x) (@norm_sq_neg_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner ring_args inner_args x)) (sub (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner x y))) (@Eq.rec.{u,0} Scalar (add (add (@normSq.{u,v} Scalar Vector inner x) (mul (@two.{u} Scalar one add) (neg (@dot.{u,v} Scalar Vector inner x y)))) (@normSq.{u,v} Scalar Vector inner y)) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (add (add (@normSq.{u,v} Scalar Vector inner x) (mul (@two.{u} Scalar one add) (neg (@dot.{u,v} Scalar Vector inner x y)))) (@normSq.{u,v} Scalar Vector inner y)) z) => @Eq.{u} Scalar (add (add (@normSq.{u,v} Scalar Vector inner x) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y)) z) (@Eq.rec.{u,0} Scalar (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y)) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y)) z) => @Eq.{u} Scalar (add (add (@normSq.{u,v} Scalar Vector inner x) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y)) (add (add (@normSq.{u,v} Scalar Vector inner x) z) (@normSq.{u,v} Scalar Vector inner y))) (@Eq.refl.{u} Scalar (add (add (@normSq.{u,v} Scalar Vector inner x) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (@normSq.{u,v} Scalar Vector inner y))) (mul (@two.{u} Scalar one add) (neg (@dot.{u,v} Scalar Vector inner x y))) (@Eq.rec.{u,0} Scalar (@dot.{u,v} Scalar Vector inner (vneg x) y) (fun (z : Scalar) => fun (hz : @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner (vneg x) y) z) => @Eq.{u} Scalar (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y)) (mul (@two.{u} Scalar one add) z)) (@Eq.refl.{u} Scalar (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (vneg x) y))) (neg (@dot.{u,v} Scalar Vector inner x y)) (@dot_neg_left_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args x y))) (sub (add (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner x y))) (@law_of_cosines_scalar_rhs_from_ring_args.{u} Scalar zero one add neg sub mul ring_args (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y) (@dot.{u,v} Scalar Vector inner x y))))

theorem dot_zero_left_from_law_packages.{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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), 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 (y : Vector), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner vzero y) zero :=
  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 ring_args => fun vector_args => fun inner_args => fun y => ring_args (@Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner vzero y) zero) (fun (sub_eq_add_neg_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (sub a b) (add a (neg b))) => fun (add_assoc_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (add (add a b) c) (add a (add b c))) => fun (add_comm_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (add a b) (add b a)) => fun (add_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (add a zero) a) => fun (zero_add_arg : forall (a : Scalar), @Eq.{u} Scalar (add zero a) a) => fun (neg_add_cancel_arg : forall (a : Scalar), @Eq.{u} Scalar (add (neg a) a) zero) => fun (add_neg_cancel_arg : forall (a : Scalar), @Eq.{u} Scalar (add a (neg a)) zero) => fun (sub_self_arg : forall (a : Scalar), @Eq.{u} Scalar (sub a a) zero) => fun (mul_assoc_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul (mul a b) c) (mul a (mul b c))) => fun (mul_comm_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (mul a b) (mul b a)) => fun (mul_one_arg : forall (a : Scalar), @Eq.{u} Scalar (mul a one) a) => fun (one_mul_arg : forall (a : Scalar), @Eq.{u} Scalar (mul one a) a) => fun (left_distrib_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul a (add b c)) (add (mul a b) (mul a c))) => fun (right_distrib_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul (add a b) c) (add (mul a c) (mul b c))) => fun (mul_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (mul a zero) zero) => fun (zero_mul_arg : forall (a : Scalar), @Eq.{u} Scalar (mul zero a) zero) => fun (add_left_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (h : @Eq.{u} Scalar (add a b) (add a c)), @Eq.{u} Scalar b c) => fun (ring_normalize_add_mul3_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (add (add (mul a b) (mul b c)) (mul a c)) (add (add (mul a b) (mul a c)) (mul b c))) => fun (add_right_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (h : @Eq.{u} Scalar (add b a) (add c a)), @Eq.{u} Scalar b c) => fun (neg_neg_arg : forall (a : Scalar), @Eq.{u} Scalar (neg (neg a)) a) => fun (sub_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (sub a zero) a) => fun (zero_sub_arg : forall (a : Scalar), @Eq.{u} Scalar (sub zero a) (neg a)) => fun (sub_add_cancel_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (add (sub a b) b) a) => fun (add_sub_cancel_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (sub (add a b) b) a) => fun (sub_add_sub_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (sub (sub a c) (sub b c)) (sub a b)) => vector_args (@Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner vzero y) zero) (fun (vec_sub_def_arg : forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x y) (vadd x (vneg y))) => fun (vec_add_assoc_arg : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (vadd (vadd x y) z) (vadd x (vadd y z))) => fun (vec_add_comm_arg : forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (vadd x y) (vadd y x)) => fun (vec_add_zero_arg : forall (x : Vector), @Eq.{v} Vector (vadd x vzero) x) => fun (vec_zero_add_arg : forall (x : Vector), @Eq.{v} Vector (vadd vzero x) x) => fun (vec_neg_add_cancel_arg : forall (x : Vector), @Eq.{v} Vector (vadd (vneg x) x) vzero) => fun (vec_add_neg_cancel_arg : forall (x : Vector), @Eq.{v} Vector (vadd x (vneg x)) vzero) => fun (sub_sub_sub_cancel_arg : 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 (vec_sub_self_arg : forall (x : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x x) vzero) => fun (vec_sub_zero_arg : forall (x : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x vzero) x) => fun (vec_add_left_cancel_arg : 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 (smul_add_arg : 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 (add_smul_arg : 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 (one_smul_arg : forall (x : Vector), @Eq.{v} Vector (smul one x) x) => fun (mul_smul_arg : forall (a : Scalar), forall (b : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (mul a b) x) (smul a (smul b x))) => fun (zero_smul_arg : forall (x : Vector), @Eq.{v} Vector (smul zero x) vzero) => fun (smul_zero_arg : forall (a : Scalar), @Eq.{v} Vector (smul a vzero) vzero) => fun (neg_smul_arg : forall (a : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (neg a) x) (vneg (smul a x))) => fun (smul_neg_arg : forall (a : Scalar), forall (x : Vector), @Eq.{v} Vector (smul a (vneg x)) (vneg (smul a x))) => fun (vec_sub_eq_add_neg_arg : forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x y) (vadd x (vneg y))) => fun (sub_add_sub_cancel_left_arg : 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 (linear_comb2_ext_arg : 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 (linear_comb3_ext_arg : 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))) => inner_args (@Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner vzero y) zero) (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 (inner_field16_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 (inner_field17_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))) => add_right_cancel_arg (@dot.{u,v} Scalar Vector inner vzero y) (@dot.{u,v} Scalar Vector inner vzero y) zero (@eq_trans.{u} Scalar (add (@dot.{u,v} Scalar Vector inner vzero y) (@dot.{u,v} Scalar Vector inner vzero y)) (@dot.{u,v} Scalar Vector inner vzero y) (add zero (@dot.{u,v} Scalar Vector inner vzero y)) (@eq_trans.{u} Scalar (add (@dot.{u,v} Scalar Vector inner vzero y) (@dot.{u,v} Scalar Vector inner vzero y)) (@dot.{u,v} Scalar Vector inner (vadd vzero vzero) y) (@dot.{u,v} Scalar Vector inner vzero y) (@eq_symm.{u} Scalar (@dot.{u,v} Scalar Vector inner (vadd vzero vzero) y) (add (@dot.{u,v} Scalar Vector inner vzero y) (@dot.{u,v} Scalar Vector inner vzero y)) (dot_add_left_arg vzero vzero y)) (@eq_congr_arg.{v,u} Vector Scalar (fun (z : Vector) => @dot.{u,v} Scalar Vector inner z y) (vadd vzero vzero) vzero (vec_zero_add_arg vzero))) (@eq_symm.{u} Scalar (add zero (@dot.{u,v} Scalar Vector inner vzero y)) (@dot.{u,v} Scalar Vector inner vzero y) (zero_add_arg (@dot.{u,v} Scalar Vector inner vzero y)))))))

theorem dot_zero_right_from_law_packages.{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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), 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 (x : Vector), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x vzero) zero :=
  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 ring_args => fun vector_args => fun inner_args => fun x => inner_args (@Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x vzero) zero) (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 (inner_field16_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 (inner_field17_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))) => @eq_trans.{u} Scalar (@dot.{u,v} Scalar Vector inner x vzero) (@dot.{u,v} Scalar Vector inner vzero x) zero (dot_comm_arg x vzero) (@dot_zero_left_from_law_packages.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner ring_args vector_args inner_args x))

theorem dot_eq_zero_of_norm_sq_zero_left_from_inner_args.{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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), 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 (x : Vector), forall (y : Vector), forall (hzero : @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner x) zero), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero :=
  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 ring_args => fun vector_args => fun inner_args => fun x => fun y => fun hzero => inner_args (@Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero) (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 (inner_field16_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 (inner_field17_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))) => @eq_trans.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) (@dot.{u,v} Scalar Vector inner vzero y) zero (@eq_congr_arg.{v,u} Vector Scalar (fun (z : Vector) => @dot.{u,v} Scalar Vector inner z y) x vzero (norm_sq_zero_iff_arg x (@Eq.{v} Vector x vzero) (fun (forward : forall (h : @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner x) zero), @Eq.{v} Vector x vzero) => fun (backward : forall (h : @Eq.{v} Vector x vzero), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner x) zero) => forward hzero))) (@dot_zero_left_from_law_packages.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner ring_args vector_args inner_args y))

theorem dot_eq_zero_of_norm_sq_zero_right_from_inner_args.{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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), 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 (x : Vector), forall (y : Vector), forall (hzero : @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner y) zero), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero :=
  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 ring_args => fun vector_args => fun inner_args => fun x => fun y => fun hzero => inner_args (@Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero) (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 (inner_field16_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 (inner_field17_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))) => @eq_trans.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) (@dot.{u,v} Scalar Vector inner x vzero) zero (@eq_congr_arg.{v,u} Vector Scalar (fun (z : Vector) => @dot.{u,v} Scalar Vector inner x z) y vzero (norm_sq_zero_iff_arg y (@Eq.{v} Vector y vzero) (fun (forward : forall (h : @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner y) zero), @Eq.{v} Vector y vzero) => fun (backward : forall (h : @Eq.{v} Vector y vzero), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner y) zero) => forward hzero))) (@dot_zero_right_from_law_packages.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner ring_args vector_args inner_args x))

theorem cauchy_schwarz_zero_left_from_law_packages.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : Scalar), Scalar), forall (ordered_args : @OrderedFieldLawArgs.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), 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 (x : Vector), forall (y : Vector), forall (hzero : @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner x) zero), le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) :=
  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 lt_rel => fun sqrt_fn => fun ordered_args => fun ring_args => fun vector_args => fun inner_args => fun x => fun y => fun hzero => ordered_args (le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) (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)))) => ring_args (le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) (fun (sub_eq_add_neg_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (sub a b) (add a (neg b))) => fun (add_assoc_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (add (add a b) c) (add a (add b c))) => fun (add_comm_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (add a b) (add b a)) => fun (add_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (add a zero) a) => fun (zero_add_arg : forall (a : Scalar), @Eq.{u} Scalar (add zero a) a) => fun (neg_add_cancel_arg : forall (a : Scalar), @Eq.{u} Scalar (add (neg a) a) zero) => fun (add_neg_cancel_arg : forall (a : Scalar), @Eq.{u} Scalar (add a (neg a)) zero) => fun (sub_self_arg : forall (a : Scalar), @Eq.{u} Scalar (sub a a) zero) => fun (mul_assoc_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul (mul a b) c) (mul a (mul b c))) => fun (mul_comm_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (mul a b) (mul b a)) => fun (mul_one_arg : forall (a : Scalar), @Eq.{u} Scalar (mul a one) a) => fun (one_mul_arg : forall (a : Scalar), @Eq.{u} Scalar (mul one a) a) => fun (left_distrib_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul a (add b c)) (add (mul a b) (mul a c))) => fun (right_distrib_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul (add a b) c) (add (mul a c) (mul b c))) => fun (mul_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (mul a zero) zero) => fun (zero_mul_arg : forall (a : Scalar), @Eq.{u} Scalar (mul zero a) zero) => fun (add_left_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (h : @Eq.{u} Scalar (add a b) (add a c)), @Eq.{u} Scalar b c) => fun (ring_normalize_add_mul3_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (add (add (mul a b) (mul b c)) (mul a c)) (add (add (mul a b) (mul a c)) (mul b c))) => fun (add_right_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (h : @Eq.{u} Scalar (add b a) (add c a)), @Eq.{u} Scalar b c) => fun (neg_neg_arg : forall (a : Scalar), @Eq.{u} Scalar (neg (neg a)) a) => fun (sub_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (sub a zero) a) => fun (zero_sub_arg : forall (a : Scalar), @Eq.{u} Scalar (sub zero a) (neg a)) => fun (sub_add_cancel_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (add (sub a b) b) a) => fun (add_sub_cancel_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (sub (add a b) b) a) => fun (sub_add_sub_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (sub (sub a c) (sub b c)) (sub a b)) => le_of_eq_arg (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (@eq_trans.{u} Scalar (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) zero (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (sq_eq_zero_iff_arg (@dot.{u,v} Scalar Vector inner x y) (@Eq.{u} Scalar (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) zero) (fun (forward : forall (hsqz : @Eq.{u} Scalar (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) zero), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero) => fun (backward : forall (haz : @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero), @Eq.{u} Scalar (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) zero) => backward (@dot_eq_zero_of_norm_sq_zero_left_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner ring_args vector_args inner_args x y hzero))) (@eq_symm.{u} Scalar (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) zero (@eq_trans.{u} Scalar (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (mul zero (@normSq.{u,v} Scalar Vector inner y)) zero (@eq_congr_arg.{u,u} Scalar Scalar (fun (z : Scalar) => mul z (@normSq.{u,v} Scalar Vector inner y)) (@normSq.{u,v} Scalar Vector inner x) zero hzero) (zero_mul_arg (@normSq.{u,v} Scalar Vector inner y))))) (le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) (fun (hab_le : le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) => fun (hba_le : le_rel (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y))) => hab_le)))

theorem cauchy_schwarz_zero_right_from_law_packages.{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 (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : Scalar), Scalar), forall (ordered_args : @OrderedFieldLawArgs.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), 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 (x : Vector), forall (y : Vector), forall (hzero : @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner y) zero), le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) :=
  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 lt_rel => fun sqrt_fn => fun ordered_args => fun ring_args => fun vector_args => fun inner_args => fun x => fun y => fun hzero => ordered_args (le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) (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)))) => ring_args (le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) (fun (sub_eq_add_neg_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (sub a b) (add a (neg b))) => fun (add_assoc_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (add (add a b) c) (add a (add b c))) => fun (add_comm_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (add a b) (add b a)) => fun (add_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (add a zero) a) => fun (zero_add_arg : forall (a : Scalar), @Eq.{u} Scalar (add zero a) a) => fun (neg_add_cancel_arg : forall (a : Scalar), @Eq.{u} Scalar (add (neg a) a) zero) => fun (add_neg_cancel_arg : forall (a : Scalar), @Eq.{u} Scalar (add a (neg a)) zero) => fun (sub_self_arg : forall (a : Scalar), @Eq.{u} Scalar (sub a a) zero) => fun (mul_assoc_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul (mul a b) c) (mul a (mul b c))) => fun (mul_comm_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (mul a b) (mul b a)) => fun (mul_one_arg : forall (a : Scalar), @Eq.{u} Scalar (mul a one) a) => fun (one_mul_arg : forall (a : Scalar), @Eq.{u} Scalar (mul one a) a) => fun (left_distrib_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul a (add b c)) (add (mul a b) (mul a c))) => fun (right_distrib_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (mul (add a b) c) (add (mul a c) (mul b c))) => fun (mul_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (mul a zero) zero) => fun (zero_mul_arg : forall (a : Scalar), @Eq.{u} Scalar (mul zero a) zero) => fun (add_left_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (h : @Eq.{u} Scalar (add a b) (add a c)), @Eq.{u} Scalar b c) => fun (ring_normalize_add_mul3_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (add (add (mul a b) (mul b c)) (mul a c)) (add (add (mul a b) (mul a c)) (mul b c))) => fun (add_right_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), forall (h : @Eq.{u} Scalar (add b a) (add c a)), @Eq.{u} Scalar b c) => fun (neg_neg_arg : forall (a : Scalar), @Eq.{u} Scalar (neg (neg a)) a) => fun (sub_zero_arg : forall (a : Scalar), @Eq.{u} Scalar (sub a zero) a) => fun (zero_sub_arg : forall (a : Scalar), @Eq.{u} Scalar (sub zero a) (neg a)) => fun (sub_add_cancel_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (add (sub a b) b) a) => fun (add_sub_cancel_arg : forall (a : Scalar), forall (b : Scalar), @Eq.{u} Scalar (sub (add a b) b) a) => fun (sub_add_sub_cancel_arg : forall (a : Scalar), forall (b : Scalar), forall (c : Scalar), @Eq.{u} Scalar (sub (sub a c) (sub b c)) (sub a b)) => le_of_eq_arg (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (@eq_trans.{u} Scalar (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) zero (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (sq_eq_zero_iff_arg (@dot.{u,v} Scalar Vector inner x y) (@Eq.{u} Scalar (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) zero) (fun (forward : forall (hsqz : @Eq.{u} Scalar (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) zero), @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero) => fun (backward : forall (haz : @Eq.{u} Scalar (@dot.{u,v} Scalar Vector inner x y) zero), @Eq.{u} Scalar (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) zero) => backward (@dot_eq_zero_of_norm_sq_zero_right_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner ring_args vector_args inner_args x y hzero))) (@eq_symm.{u} Scalar (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) zero (@eq_trans.{u} Scalar (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (mul (@normSq.{u,v} Scalar Vector inner x) zero) zero (@eq_congr_arg.{u,u} Scalar Scalar (fun (z : Scalar) => mul (@normSq.{u,v} Scalar Vector inner x) z) (@normSq.{u,v} Scalar Vector inner y) zero hzero) (mul_zero_arg (@normSq.{u,v} Scalar Vector inner x))))) (le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) (fun (hab_le : le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) => fun (hba_le : le_rel (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y))) => hab_le)))

theorem cauchy_schwarz_from_law_packages.{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 (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 (x : Vector), forall (y : Vector), le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y)) :=
  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 ring_args => fun ordered_args => fun vector_args => fun inner_args => fun x => fun y => inner_args (le_rel (@sq.{u} Scalar mul (@dot.{u,v} Scalar Vector inner x y)) (mul (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y))) (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 (inner_field16_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 (inner_field17_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))) => @square_completion_bound_from_ordered_args.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn ordered_args (@normSq.{u,v} Scalar Vector inner x) (@dot.{u,v} Scalar Vector inner x y) (@normSq.{u,v} Scalar Vector inner y) (quadratic_norm_nonneg_arg x y))

theorem norm_sq_nonneg_from_inner_args.{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 (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (x : Vector), le_rel zero (@normSq.{u,v} Scalar Vector inner x) :=
  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 inner_args => fun x => inner_args (le_rel zero (@normSq.{u,v} Scalar Vector inner x)) (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 (inner_field16_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 (inner_field17_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 x)

theorem dot_le_mul_sqrt_norm_sq_from_cauchy.{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 (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 (x : Vector), forall (y : Vector), le_rel (@dot.{u,v} Scalar Vector inner x y) (mul (sqrt_fn (@normSq.{u,v} Scalar Vector inner x)) (sqrt_fn (@normSq.{u,v} Scalar Vector inner y))) :=
  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 ring_args => fun ordered_args => fun vector_args => fun inner_args => fun x => fun y => @le_mul_sqrt_of_sq_le_mul_nonneg_from_ordered_args.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn ordered_args (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y) (@dot.{u,v} Scalar Vector inner x y) (@norm_sq_nonneg_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args x) (@norm_sq_nonneg_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args y) (@cauchy_schwarz_from_law_packages.{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 x y)

theorem norm_sq_add_le_square_sum_norms_from_cauchy.{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 (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 (x : Vector), forall (y : Vector), le_rel (@normSq.{u,v} Scalar Vector inner (vadd x y)) (@sq.{u} Scalar mul (add (sqrt_fn (@normSq.{u,v} Scalar Vector inner x)) (sqrt_fn (@normSq.{u,v} Scalar Vector inner y)))) :=
  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 ring_args => fun ordered_args => fun vector_args => fun inner_args => fun x => fun y => @add_two_mul_le_sq_add_sqrt_from_ordered_args.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn ordered_args (@normSq.{u,v} Scalar Vector inner x) (@normSq.{u,v} Scalar Vector inner y) (@dot.{u,v} Scalar Vector inner x y) (@normSq.{u,v} Scalar Vector inner (vadd x y)) (@norm_sq_nonneg_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args x) (@norm_sq_nonneg_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args y) (@norm_sq_add_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner inner_args x y) (@dot_le_mul_sqrt_norm_sq_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 x y)