parallelogram_law

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 (law : 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)))), 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 law => fun x => fun y => law x y