dot_neg_left_from_inner_args

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)