## Notations

We denote :

- Pressure : ν ∈ [ 0 , 1 ] \nu \in \lbrack 0,1\rbrack

- time : t ≥ 0 t \geq 0

- point : defined by position x x and y y (or p = ( x , y ) ∈ ℝ 2 p = (x,y) \in {\mathbb{R}}^{2} )

- Raw inputs are denoted by a tuple i [ k ] = ( ν [ k ] , t [ k ] , x [ k ] , y [ k ] ) i\lbrack k\rbrack = \left( \nu\lbrack k\rbrack,t\lbrack k\rbrack,x\lbrack k\rbrack,y\lbrack k\rbrack \right) with k ∈ ℕ k \in {\mathbb{N}} .

An input stream is a sequence of raw inputs i = ( ν , t , x , y ) i = (\nu,t,x,y)

- with time t [ k ] t\lbrack k\rbrack ↗ ↗ \nearrow \nearrow strictly increasing

- starts with a `Down` event

- contains `Move` for k ≥ 1 k \geq 1

- ends either with a `Move` or a `Up` . If it is a `Up` we say the input stream is complete

We addition define

- v x , v y , a x , a y v_{x},v_{y},a_{x},a_{y} as the velocity and acceleration.

With the vector shorthand v = ( v x , v y ) ∈ ℝ 2 v = \left( v_{x},v_{y} \right) \in {\mathbb{R}}^{2} and a = ( a x , a y ) ∈ ℝ 2 a = \left( a_{x},a_{y} \right) \in {\mathbb{R}}^{2}