Fact.cs

    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public OnLineFact(OnLineFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Pid], old_to_new[fact.Rid]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="pid">sets <see cref="Pid"/></param>
    /// <param name="rid">sets <see cref="Rid"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public OnLineFact(string pid, string rid, FactOrganizer organizer) : base(organizer)
        => init(pid, rid);

    /// <summary>
    /// Initiates <see cref="Pid"/>, <see cref="Rid"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="pid">sets <see cref="Pid"/></param>
    /// <param name="rid">sets <see cref="Rid"/></param>
    private void init(string pid, string rid)
    {
        this.Pid = pid;
        this.Rid = rid;

        MMTTerm tp = new OMA(
            new OMS(MMTURIs.Ded), 
            new List<MMTTerm> {
                new OMA(
                    new OMS(MMTURIs.OnLine),
                    new List<MMTTerm> {
                        new OMS(rid),
                        new OMS(pid)
        }),});

        MMTTerm df = null;

        MMTSymbolDeclaration mmtDecl = new (this.Label, tp, df);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="pid">sets <see cref="Pid"/></param>
    /// <param name="rid">sets <see cref="Rid"/></param>
    /// <param name="uri">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public OnLineFact(string pid, string rid, string uri, FactOrganizer organizer) : base(organizer)
    {
        this.Pid = pid;
        this.Rid = rid;
        this._URI = uri;
        _ = this.Label;
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static OnLineFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;

        string pointUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;
        string lineUri = ((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0] is OMS
            // standard case
            ? ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri
            // case when line Uri has a projl on the line Argument 
            : ((OMS)((OMA)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).arguments[0]).uri;

        if (StageStatic.stage.factState.ContainsKey(pointUri)
         && StageStatic.stage.factState.ContainsKey(lineUri))
            return new OnLineFact(pointUri, lineUri, uri, StageStatic.stage.factState);

        //If dependent facts do not exist return null
        else return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => _Facts[Pid].Label + "∈" + _Facts[Rid].Label;

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Pid, Rid };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Pid].Label;
        obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Rid].Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    public override int GetHashCode()
        => this.Pid.GetHashCode() ^ this.Rid.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(OnLineFact f1, OnLineFact f2)
    {
        if (f1.Pid == f2.Pid && f1.Rid == f2.Rid)
            return true;

        PointFact pf1 = (PointFact)_Facts[f1.Pid];
        RayFact rf1 = (RayFact)_Facts[f1.Rid];
        PointFact pf2 = (PointFact)_Facts[f2.Pid];
        RayFact rf2 = (RayFact)_Facts[f2.Rid];

        return pf1.Equivalent(pf2) && rf1.Equivalent(rf2);
    }
}

/// <summary>
/// Angle comprised of three <see cref="PointFact">PointFacts</see> [A,B,C]
/// </summary>
public class AngleFact : FactWrappedCRTP<AngleFact>
{
    /// \copydoc Fact.s_type
    [JsonProperty]
    protected static new string s_type = "AngleFact";

    /// @{ <summary>
    /// One <see cref="Fact.Id">Id</see> of three <see cref="PointFact">PointFacts</see> defining Angle [<see cref="Pid1"/>, <see cref="Pid2"/>, <see cref="Pid3"/>].
    /// </summary>
    public string Pid1, Pid2, Pid3;
    /// @}

    /// <summary> <see langword="true"/>, if AngleFact is approximately 90° or 270°</summary>
    public bool is_right_angle;


    /// <summary> \copydoc Fact.Fact </summary>
    public AngleFact() : base()
    {
        this.Pid1 = null;
        this.Pid2 = null;
        this.Pid3 = null;
        this.is_right_angle = false;
    }

    /// <summary>
    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public AngleFact(AngleFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Pid1], old_to_new[fact.Pid2], old_to_new[fact.Pid3]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="pid2">sets <see cref="Pid2"/></param>
    /// <param name="pid3">sets <see cref="Pid3"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public AngleFact(string pid1, string pid2, string pid3, FactOrganizer organizer) : base(organizer)
        => init(pid1, pid2, pid3);

    /// <summary>
    /// Initiates <see cref="Pid1"/>, <see cref="Pid2"/>, <see cref="Pid3"/>, <see cref="is_right_angle"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="pid2">sets <see cref="Pid2"/></param>
    /// <param name="pid3">sets <see cref="Pid3"/></param>
    private void init(string pid1, string pid2, string pid3)
    {
        this.Pid1 = pid1;
        this.Pid2 = pid2;
        this.Pid3 = pid3;

        float v = GetAngle(); // sets is_right_angle

        MMTDeclaration mmtDecl = generateNot90DegreeAngleDeclaration(v, pid1, pid2, pid3);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Pid1">sets <see cref="Pid1"/></param>
    /// <param name="Pid2">sets <see cref="Pid2"/></param>
    /// <param name="Pid3">sets <see cref="Pid3"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public AngleFact(string Pid1, string Pid2, string Pid3, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Pid1 = Pid1;
        this.Pid2 = Pid2;
        this.Pid3 = Pid3;

        _ = GetAngle();
        this._URI = backendURI;
        _ = this.Label;
    }

    public AngleFact(string Pid1, string Pid2, string Pid3, float angle, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Pid1 = Pid1;
        this.Pid2 = Pid2;
        this.Pid3 = Pid3;

        this._URI = backendURI;
        _ = this.Label;
    }


    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static AngleFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;
        string
            pointAUri,
            pointBUri,
            pointCUri;

        float angle = 0.0f;
        //If angle is not a 90Degree-Angle
        if (fact.GetType().Equals(typeof(Scroll.ScrollValueFact)))
        {
            OMA df = (OMA)((Scroll.ScrollValueFact)fact).lhs;
            if (df == null)
                return null;

            if (((Scroll.ScrollValueFact)fact).value != null)
                angle = ((OMF)(((Scroll.ScrollValueFact)fact).value)).f;

            pointAUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).uri;
            pointBUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[1]).uri;
            pointCUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[2]).uri;
        }
        // this should never happen anymore
        //If angle is a 90Degree-Angle
        else
        {
            Debug.LogError("Angle 90 degrees parsed. This shouldn't happen anymore");

            pointAUri = ((OMS)((OMA)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).arguments[0]).uri;
            pointBUri = ((OMS)((OMA)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).arguments[1]).uri;
            pointCUri = ((OMS)((OMA)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).arguments[2]).uri;
        }

        if (StageStatic.stage.factState.ContainsKey(pointAUri)
         && StageStatic.stage.factState.ContainsKey(pointBUri)
         && StageStatic.stage.factState.ContainsKey(pointCUri))
            return new AngleFact(pointAUri, pointBUri, pointCUri, angle, uri, StageStatic.stage.factState);
        else
            return null; //If dependent facts do not exist return null
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => (is_right_angle ? "⊾" : "∠") + _Facts[Pid1].Label + _Facts[Pid2].Label + _Facts[Pid3].Label;

    /// <summary>
    /// Computes smallest angle and sets <see cref="is_right_angle"/>
    /// </summary>
    /// <returns>Smallets angle between [<see cref="Pid1"/>, <see cref="Pid2"/>] and [<see cref="Pid2"/>, <see cref="Pid3"/>]</returns>
    private float GetAngle()
    {
        PointFact pf1 = _Facts[Pid1] as PointFact;
        PointFact pf2 = _Facts[Pid2] as PointFact;
        PointFact pf3 = _Facts[Pid3] as PointFact;

        float v = Vector3.Angle((pf1.Point - pf2.Point), (pf3.Point - pf2.Point));
        this.is_right_angle = Mathf.Abs(v - 90.0f) < 0.01;

        return is_right_angle ? 90f : v;
    }

    /// <summary>
    /// Constructs struct for right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="val">Angle == 90f, _not checked_</param>
    /// <param name="p1URI"><see cref="Pid1"/></param>
    /// <param name="p2URI"><see cref="Pid2"/></param>
    /// <param name="p3URI"><see cref="Pid3"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generate90DegreeAngleDeclaration(float val, string p1URI, string p2URI, string p3URI)
    {
        MMTTerm argument = new OMA(
            new OMS(MMTURIs.Eq),
            new List<MMTTerm> {
                new OMS(MMTURIs.RealLit),
                new OMA(
                    new OMS(MMTURIs.Angle),
                    new List<MMTTerm> {
                        new OMS(p1URI),
                        new OMS(p2URI),
                        new OMS(p3URI)
                    }
                ),
                new OMF(val) // 90f
            }
        );

        MMTTerm tp = new OMA(new OMS(MMTURIs.Ded), new List<MMTTerm> { argument });
        MMTTerm df = null;

        return new MMTSymbolDeclaration(this.Label, tp, df);
    }

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="val">Angle != 90f, _not checked_</param>
    /// <param name="p1URI"><see cref="Pid1"/></param>
    /// <param name="p2URI"><see cref="Pid2"/></param>
    /// <param name="p3URI"><see cref="Pid3"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateNot90DegreeAngleDeclaration(float val, string p1URI, string p2URI, string p3URI)
    {
        MMTTerm lhs =
            new OMA(
                new OMS(MMTURIs.Angle),
                new List<MMTTerm> {
                    new OMS(p1URI),
                    new OMS(p2URI),
                    new OMS(p3URI)
                }
            );

        MMTTerm valueTp = new OMS(MMTURIs.RealLit);
        MMTTerm value = new OMF(val);

        return new MMTValueDeclaration(this.Label, lhs, valueTp, value);
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Pid1, Pid2, Pid3 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Pid1].Label;
        obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Pid2].Label;
        obj.transform.GetChild(2).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Pid3].Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    public override int GetHashCode()
        => this.Pid1.GetHashCode() ^ this.Pid2.GetHashCode() ^ this.Pid3.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(AngleFact f1, AngleFact f2)
    {
        if ((f1.Pid1 == f2.Pid1 && f1.Pid2 == f2.Pid2 && f1.Pid3 == f2.Pid3))// || 
            //(f1.Pid1 == f2.Pid3 && f1.Pid2 == f2.Pid2 && f1.Pid3 == f2.Pid1))
            return true;

        PointFact p1f1 = (PointFact)_Facts[f1.Pid1];
        PointFact p2f1 = (PointFact)_Facts[f1.Pid2];
        PointFact p3f1 = (PointFact)_Facts[f1.Pid3];
        PointFact p1f2 = (PointFact)_Facts[f2.Pid1];
        PointFact p2f2 = (PointFact)_Facts[f2.Pid2];
        PointFact p3f2 = (PointFact)_Facts[f2.Pid3];

        return (p1f1.Equivalent(p1f2) && p2f1.Equivalent(p2f2) && p3f1.Equivalent(p3f2));
        //|| (p1f1.Equivalent(p3f2) && p2f1.Equivalent(p2f2) && p1f1.Equivalent(p3f2));
    }
}

/// <summary>
/// A RightAngleFact defined by 3  <see cref="PointFact">Pointfact</see> 
/// </summary>
public class RightAngleFact : FactWrappedCRTP<RightAngleFact>
{
    /// \copydoc Fact.s_type
    [JsonProperty]
    protected static new string s_type = "RightAngleFact";

    ///  <summary> three <see cref="PointFact">Pointfacts</see> defining the right angle </summary>
    public string Pid1, Pid2, Pid3;

    /// <summary> \copydoc Fact.Fact </summary>
    public RightAngleFact() : base()
    {
        this.Pid1 = null;
        this.Pid2 = null;
        this.Pid3 = null;
    }

    /// <summary>
    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public RightAngleFact(RightAngleFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Pid1], old_to_new[fact.Pid2], old_to_new[fact.Pid3]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="pid2">sets <see cref="Pid2"/></param>
    /// <param name="pid3">sets <see cref="Pid3"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public RightAngleFact(string pid1, string pid2, string pid3, FactOrganizer organizer) : base(organizer)
        => init(pid1, pid2, pid3);

    /// <summary>
    /// Initiates <see cref="Pid1"/>, <see cref="Pid2"/>, <see cref="Pid3"/>, <see cref="is_right_angle"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="pid2">sets <see cref="Pid2"/></param>
    /// <param name="pid3">sets <see cref="Pid3"/></param>
    private void init(string pid1, string pid2, string pid3)
    {
        this.Pid1 = pid1;
        this.Pid2 = pid2;
        this.Pid3 = pid3;

        AddFactResponse.sendAdd(generateMMTDeclaration(pid1, pid2, pid3), out this._URI);
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Pid1">sets <see cref="Pid1"/></param>
    /// <param name="Pid2">sets <see cref="Pid2"/></param>
    /// <param name="Pid3">sets <see cref="Pid3"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public RightAngleFact(string Pid1, string Pid2, string Pid3, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Pid1 = Pid1;
        this.Pid2 = Pid2;
        this.Pid3 = Pid3;

        this._URI = backendURI;
        _ = this.Label;
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static RightAngleFact parseFact(Scroll.ScrollFact fact)
    {
        OMA tp = (OMA)((Scroll.ScrollSymbolFact)fact).tp;
        if (tp == null)
            return null;

        string Point1Uri = "";
        string Point2Uri = "";
        string Point3Uri = "";

        string uri = fact.@ref.uri;
        OMA proof_OMA = (OMA)((Scroll.ScrollSymbolFact)fact).tp; // proof DED
        OMA rightAngleOMA = (OMA)proof_OMA.arguments[0]; // rightAngle OMA

        if (rightAngleOMA.arguments[0] is OMS)
        {
            Point1Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
            Point2Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;
            Point3Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[2]).uri;
        }

        if (StageStatic.stage.factState.ContainsKey(Point1Uri)
         && StageStatic.stage.factState.ContainsKey(Point2Uri)
         && StageStatic.stage.factState.ContainsKey(Point3Uri))

            return new RightAngleFact(Point1Uri, Point2Uri, Point3Uri, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }


    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => _Facts[Pid1].Label + _Facts[Pid2].Label + _Facts[Pid3].Label + "⊥";


    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="p1URI"> Uri for <see cref="Pid1"/></param>
    /// <param name="p2URI"> Uri for <see cref="Pid2"/></param>
    /// <param name="p3URI"> Uri for <see cref="Pid3"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string p1URI, string p2URI, string p3URI)
    {
        MMTTerm tp =
            new OMA(
                new OMS(MMTURIs.Ded),
                new List<MMTTerm> {
                    new OMA(
                        new OMS(MMTURIs.RightAngle),
                        new List<MMTTerm> {
                            new OMS(p1URI),
                            new OMS(p2URI),
                            new OMS(p3URI),
        }),});

        MMTTerm df = null;

        return new MMTSymbolDeclaration(this.Label, tp, df);
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Pid1, Pid2, Pid3 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Pid1].Label;
        obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Pid2].Label;
        obj.transform.GetChild(2).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Pid3].Label;

        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    /// uhhh is this a problem?
    public override int GetHashCode()
        => this.Pid1.GetHashCode() ^ this.Pid2.GetHashCode() ^ this.Pid3.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(RightAngleFact f1, RightAngleFact f2)
    {
        if (f1.Pid1 == f2.Pid1 && f1.Pid2 == f2.Pid2 && f1.Pid3 == f2.Pid3)
            return true;

        PointFact p1f1 = (PointFact)_Facts[f1.Pid1];
        PointFact p2f1 = (PointFact)_Facts[f1.Pid2];
        PointFact p3f1 = (PointFact)_Facts[f1.Pid3];

        PointFact p1f2 = (PointFact)_Facts[f2.Pid1];
        PointFact p2f2 = (PointFact)_Facts[f2.Pid2];
        PointFact p3f2 = (PointFact)_Facts[f2.Pid3];

        return (p1f1.Equivalent(p1f2) && p2f1.Equivalent(p2f2) && p3f1.Equivalent(p3f2));
    }
}

/// <summary>
/// Two parallel Lines comprised of two <see cref="LineFact">LineFacts</see> 
/// </summary>
public class ParallelLineFact : FactWrappedCRTP<ParallelLineFact>
{
    /// \copydoc Fact.s_type
    [JsonProperty]
    protected static new string s_type = "ParallelLineFact";

    /// @{ <summary>
    /// One <see cref="Fact.Id">Id</see> of thwo <see cref="LineFact">PointFacts</see> defining Angle [<see cref="Lid1"/>, <see cref="Lid2"/>].
    /// </summary>
    public string Lid1, Lid2;
    /// @}


    /// <summary> \copydoc Fact.Fact </summary>
    public ParallelLineFact() : base()
    {
        this.Lid1 = null;
        this.Lid2 = null;
    }

    /// <summary>
    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public ParallelLineFact(ParallelLineFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
    {
        init(old_to_new[fact.Lid1], old_to_new[fact.Lid2]);
    }

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="lid1">sets <see cref="Lid1"/></param>
    /// <param name="lid2">sets <see cref="Lid2"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public ParallelLineFact(string lid1, string lid2, FactOrganizer organizer) : base(organizer)
    {
        init(lid1, lid2);
    }

    /// <summary>
    /// Initiates <see cref="Lid1"/>, <see cref="Lid2"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="lid1">sets <see cref="Lid1"/></param>
    /// <param name="lid2">sets <see cref="Lid2"/></param>
    private void init(string lid1, string lid2)
    {

        this.Lid1 = lid1;
        this.Lid2 = lid2;

        RayFact lf1 = _Facts[lid1] as RayFact;
        RayFact lf2 = _Facts[lid2] as RayFact;

        MMTDeclaration mmtDecl;
        string l1URI = lf1.Id;
        string l2URI = lf2.Id;
        mmtDecl = generateParallelLineDeclaration(l1URI, l2URI);

        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Lid1">sets <see cref="Lid1"/></param>
    /// <param name="Lid2">sets <see cref="Lid2"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public ParallelLineFact(string Lid1, string Lid2, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Lid1 = Lid1;
        this.Lid2 = Lid2;

        this._URI = backendURI;
        _ = this.Label;
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static ParallelLineFact parseFact(Scroll.ScrollFact fact)
    {
        OMA tp = (OMA)((Scroll.ScrollSymbolFact)fact).tp;
        if (tp == null)
            return null;


        string uri = fact.@ref.uri;

        string lineAUri;
        string lineBUri;

        OMA proof_OMA = (OMA)((Scroll.ScrollSymbolFact)fact).tp; // proof DED
        OMA parallel_lines_OMA = (OMA)proof_OMA.arguments[0]; // parallel

        if (parallel_lines_OMA.arguments[0] is OMS)
        {   // Normaler Fall 
            lineAUri = ((OMS) parallel_lines_OMA.arguments[0]).uri;
            lineBUri = ((OMS) parallel_lines_OMA.arguments[1]).uri;
        }
        else // TODO: Second case might be redundant by now
        {
            OMA Projl_line_A_OMA = (OMA)parallel_lines_OMA.arguments[0]; // ProjectL
            lineAUri = ((OMS)Projl_line_A_OMA.arguments[0]).uri;

            OMA Projl_line_B_OMA = (OMA)parallel_lines_OMA.arguments[1]; // ProjectL
            lineBUri = ((OMS)Projl_line_B_OMA.arguments[0]).uri;
        }


        if (StageStatic.stage.factState.ContainsKey(lineAUri)
         && StageStatic.stage.factState.ContainsKey(lineBUri))

            return new ParallelLineFact(lineAUri, lineBUri, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel() 
        => _Facts[Lid1].Label + "||" + _Facts[Lid2].Label;

    /// <summary>
    /// Constructs struct for right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="l1URI"><see cref="Lid1"/></param>
    /// <param name="l2URI"><see cref="Lid2"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateParallelLineDeclaration(string l1URI, string l2URI)
    {
        MMTTerm tp = new OMA(
            new OMS(MMTURIs.Ded),
            new List<MMTTerm> {
                new OMA(
                    new OMS(MMTURIs.ParallelLine),
                    new List<MMTTerm> {
                        new OMS(l1URI),
                        new OMS(l2URI),
                    }
                ),
            }
        );

        MMTTerm df = null;

        MMTSymbolDeclaration mmtDecl = new MMTSymbolDeclaration(this.Label, tp, df);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
        return mmtDecl;
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts() 
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds() 
        => new string[] { Lid1, Lid2 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Lid1].Label;
        obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Lid2].Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    public override int GetHashCode()
        => this.Lid1.GetHashCode() ^ this.Lid2.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(ParallelLineFact f1, ParallelLineFact f2)
    {
        if ((f1.Lid1 == f2.Lid1 && f1.Lid2 == f2.Lid2))
            return true;

        RayFact r1f1 = (RayFact)_Facts[f1.Lid1];
        RayFact r2f1 = (RayFact)_Facts[f1.Lid2];
        RayFact r1f2 = (RayFact)_Facts[f2.Lid1];
        RayFact r2f2 = (RayFact)_Facts[f2.Lid2];

        return (r1f1.Equivalent(r1f2) && r2f1.Equivalent(r2f2));
    }
}

/// <summary>
/// A Circle that is made out of a middle point, a plane and a radius
/// </summary>
public class CircleFact : FactWrappedCRTP<CircleFact>
{
    /// \copydoc Fact.s_type
    [JsonProperty]
    protected static new string s_type = "CircleFact";

    /// <summary> defining the middle point of the circle  </summary>
    public string Pid1;
    /// <summary>  defining the base point of the circle plane </summary>
    public string Pid2;
    /// <summary>  radius of the circle </summary>
    public float radius;
    /// <summary> normal vector of the plane </summary>
    public Vector3 normal;

    /// <summary> \copydoc Fact.Fact </summary>
    public CircleFact() : base()
    {
        this.normal = Vector3.zero;
        this.Pid1 = null;
        this.Pid2 = null;
        this.radius = 0;
    }

    /// <summary>
    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public CircleFact(CircleFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Pid1], old_to_new[fact.Pid2], fact.radius, fact.normal);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="pid2">sets <see cref="Pid2"/></param>
    /// <param name="radius">sets <see cref="radius"/></param>
    /// <param name="normal">sets <see cref="normal"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public CircleFact(string pid1, string pid2, float radius, Vector3 normal, FactOrganizer organizer) : base(organizer)
        => init(pid1, pid2, radius, normal);

    /// <summary>
    /// Initiates <see cref="Pid1"/>, <see cref="Pid2"/>, <see cref="radius"/>,<see cref="dir1"/>,<see cref="dir2"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="pid2">sets <see cref="Pid2"/></param>
    /// <param name="radius">sets <see cref="radius"/></param>
    /// <param name="normal">sets <see cref="normal"/></param>
    private void init(string pid1, string pid2, float radius, Vector3 normal)
    {
        this.Pid1 = pid1;
        this.Pid2 = pid2;

        this.radius = radius;
        this.normal = normal;

        MMTDeclaration mmtDecl = generateCircleFactDeclaration(pid1, pid2, radius, normal);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Pid1">sets <see cref="Pid1"/></param>
    /// <param name="Pid2">sets <see cref="Pid2"/></param>
    /// <param name="radius">sets <see cref="radius"/></param>
    /// <param name="normal">sets <see cref="normal"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public CircleFact(string Pid1, string Pid2, float radius, Vector3 normal, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Pid1 = Pid1;
        this.Pid2 = Pid2;

        this.radius = radius;
        this.normal = normal;

        this._URI = backendURI;
        _ = this.Label;
    }

    /// <summary>
    /// parses the Circlefact response of the MMT-Server
    /// </summary>
    /// \copydoc Fact.parseFact(Scroll.ScrollFact) 
    public new static CircleFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;

        OMA df = (OMA)((Scroll.ScrollSymbolFact)fact).df;
        if (df == null)
            return null;

        Scroll.ScrollSymbolFact scoll_symbol_fact = (Scroll.ScrollSymbolFact)fact;

        // get the radius
        float radius = ((OMF)((OMA)scoll_symbol_fact.df).arguments[2]).f;

        OMA planeOMA = (OMA)((OMA)scoll_symbol_fact.df).arguments[0];
        string planeApplicant = ((OMS)planeOMA.applicant).uri;

        string A_uri;
        Vector3 normal;
        // Getting the plane
        // IN case of a normale plane
        if (planeApplicant.Equals(MMTURIs.pointNormalPlane))
        {
            //OMA pointAOMA = (OMA)planeOMA.arguments[0];
            A_uri = ParsingDictionary.parseTermsToId[planeOMA.arguments[0].ToString()];

            OMA n = (OMA)planeOMA.arguments[1];
            normal = new Vector3(((OMF)n.arguments[0]).f, ((OMF)n.arguments[1]).f, ((OMF)n.arguments[2]).f);
        }
        // In case of parametrized plane
        else if (planeApplicant.Equals(MMTURIs.ParametrizedPlane))
        {
            A_uri = ParsingDictionary.parseTermsToId[planeOMA.arguments[0].ToString()];

            OMA vOMA = (OMA)planeOMA.arguments[1];
            OMA wOMA = (OMA)planeOMA.arguments[2];

            Vector3 v = new Vector3(((OMF)vOMA.arguments[0]).f, ((OMF)vOMA.arguments[1]).f, ((OMF)vOMA.arguments[2]).f);
            Vector3 w = new Vector3(((OMF)wOMA.arguments[0]).f, ((OMF)wOMA.arguments[1]).f, ((OMF)wOMA.arguments[2]).f);

            normal = Vector3.Cross(v, w).normalized;
        }
        // incase of smth else. Shouldn't hapepen unless there is an error
        else throw new ArgumentException("Invalid planeApplicant: " + planeApplicant);

        // get the mid point uri
        string parse_id_M = ((OMA)scoll_symbol_fact.df).arguments[1].ToString();
        string M_uri = ParsingDictionary.parseTermsToId[parse_id_M];

        if (StageStatic.stage.factState.ContainsKey(M_uri)
         && StageStatic.stage.factState.ContainsKey(A_uri))
            return new CircleFact(M_uri, A_uri, radius, normal, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "○" + _Facts[Pid1].Label;

    /// <summary>
    /// Constructs struct for right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="p1URI"> <see cref="Pid1"/></param>
    /// <param name="p2URI"> <see cref="Pid2"/></param>
    /// <param name="radius"> <see cref="radius"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateCircleFactDeclaration(string p1URI, string p2URI, float radius, Vector3 normal)
    {
        List<MMTTerm> outerArguments = new List<MMTTerm>
        {
           //CirclePlane,
           new OMA(
               //PointNormalPlane,
               new OMS(MMTURIs.pointNormalPlane), 
               //planeArgs,
               new List<MMTTerm> {
                    //base point of the circle plane?,
                    new OMS(p2URI),
                    //NormalVector,
                    new OMA(
                        //"Vector"
                        new OMS(MMTURIs.Tuple),
                        //normalArgs,
                        new List<MMTTerm> {
                            new OMF(normal.x),
                            new OMF(normal.y),
                            new OMF(normal.z)
                        }
                    ),
                }
            ),
           //middlePoint,
           new OMS(p1URI),
           //Radius,
           new OMF(radius),
        };

        // Do i need this here? doubt 
        MMTTerm tp = new OMS(MMTURIs.CircleType3d);
        MMTTerm df = new OMA(new OMS(MMTURIs.MkCircle3d), outerArguments);

        MMTSymbolDeclaration mmtDecl = new MMTSymbolDeclaration(Label, tp, df);
        AddFactResponse.sendAdd(mmtDecl, out _URI);

        return mmtDecl;
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Pid1, Pid2 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Pid1].Label;

        // obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Lid2].Label;

        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    public override int GetHashCode()
        => Pid1.GetHashCode() ^ Pid2.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(CircleFact f1, CircleFact f2)
    {
        if (f1.Pid1 == f2.Pid1 && f1.normal == f2.normal && f1.radius == f2.radius)
            return true;

        PointFact p1f1 = (PointFact)_Facts[f1.Pid1];
        PointFact p1f2 = (PointFact)_Facts[f2.Pid1];

        return p1f1.Equivalent(p1f2)
            && Math3d.IsApproximatelyEqual(f1.normal, f2.normal)
            && Mathf.Approximately(f1.radius, f2.radius);
    }
}

/// <summary>
/// A <see cref="PointFact"/> on a <see cref="CircleFact"/>
/// </summary>
public class OnCircleFact : FactWrappedCRTP<OnCircleFact>
{
    /// \copydoc Fact.s_type
    [JsonProperty]
    protected static new string s_type = "OnCircleFact";

    /// <summary> the point on the circle  </summary>
    public string Pid;
    /// <summary> the circle, which the point is on  </summary>
    public string Cid;