dot_zero_left_from_law_packages

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)))))))