fact: Definition cross_first_coordinate (e1 e2 : edge) := let yl2 := p_y (left_pt e2) in let xl2 := p_x (left_pt e2) in let yl1 := p_y (left_pt e1) in let xl1 := p_x (left_pt e1) in let yr2 := p_y (right_pt e2) in let xr2 := p_x (right_pt e2) in let yr1 := p_y (right_pt e1) in let xr1 := p_x (right_pt e1) in let x1 := xr1 - xl1 in let x2 := xr2 - xl2 in let y1 := yr1 - yl1 in let y2 := yr2 - yl2 in Qred ( (yl2 * x1 * x2 - yl1 * x1 * x2 - x1 * xl2 * y2 + xl1 * y1 * x2) / (y1 * x2 - x1 * y2)).
imports: Arith QArith List smooth_trajectories Reals