Fact.cs

    /// <summary> \copydoc Fact.Fact </summary>
    public OnCircleFact() : base()
    {
        this.Pid = null;
        this.Cid = 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 OnCircleFact(OnCircleFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Pid], old_to_new[fact.Cid]);

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

    /// <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="cid">sets <see cref="Cid"/></param>
    private void init(string pid, string cid)
    {
        this.Pid = pid;
        this.Cid = cid;;

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

        MMTTerm df = null;

        AddFactResponse.sendAdd(new MMTSymbolDeclaration(Label, tp, df), out _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="cid">sets <see cref="Cid"/></param>
    /// <param name="uri">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public OnCircleFact(string pid, string cid, string uri, FactOrganizer organizer) : base(organizer)
    {
        this.Pid = pid;
        this.Cid = cid;
        this._URI = uri;
        _ = this.Label;
    }

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

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

        string circleUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
        string pointUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;

        if (StageStatic.stage.factState.ContainsKey(pointUri)
         && StageStatic.stage.factState.ContainsKey(circleUri))
            return new OnCircleFact(pointUri, circleUri, 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[Cid].Label;

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

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

    /// \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 + "∈" + _Facts[this.Cid].Label;
        obj.GetComponent<FactWrapper>().fact = this;

        return obj;
    }

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

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(OnCircleFact c1, OnCircleFact c2)
    {
        if (c1.Pid == c2.Pid && c1.Cid == c2.Pid)
            return true;

        PointFact pc1 = (PointFact)_Facts[c1.Pid];
        CircleFact cc1 = (CircleFact)_Facts[c1.Cid];

        PointFact pc2 = (PointFact)_Facts[c2.Pid];
        CircleFact cc2 = (CircleFact)_Facts[c2.Cid];

        return pc1.Equivalent(pc2) && cc1.Equivalent(cc2);
    }
}

/// <summary>
/// Angle comprised of a line and a circle 
/// </summary>
public class AngleCircleLineFact : FactWrappedCRTP<AngleCircleLineFact>
{
    /// \copydoc Fact.s_type
    [JsonProperty]
    protected static new string s_type = "AngleCircleLineFact";

    /// @{ <summary>
    /// One <see cref="Fact.Id">Id</see> of a <see cref="RayFact">RayFact</see> and a <see cref="CircleFact">CircleFact</see>  defining Angle [<see cref="Cid1"/>, <see cref="Rid2"/>].
    /// </summary>
    public string Cid1, Rid2;
    /// @}
    public float angle;

    /// <summary> \copydoc Fact.Fact </summary>
    public AngleCircleLineFact() : base()
    {
        this.Cid1 = null;
        this.Rid2 = null;
        this.angle = 0.0f;
    }

    /// <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 AngleCircleLineFact(AngleCircleLineFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Cid1], old_to_new[fact.Rid2], fact.angle);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="rid2">sets <see cref="Rid2"/></param>
    /// <param name="angle"> sets the angle </param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public AngleCircleLineFact(string cid1, string rid2, float angle, FactOrganizer organizer) : base(organizer)
        => init(cid1, rid2, angle);

    /// <summary>
    /// Initiates <see cref="Cid1"/>, <see cref="Rid2"/>, <see cref="angle"/> <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="rid2">sets <see cref="Rid2"/></param>
    /// <param name="angle"> sets the angle </param>
    private void init(string cid1, string rid2, float angle)
    {
        this.Cid1 = cid1;
        this.Rid2 = rid2;
        this.angle = angle;

        MMTDeclaration mmtDecl = generateMMTDeclaration(angle, cid1, rid2);
        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="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="´Rid2">sets <see cref="Rid2"/></param>
    /// <param name="angle"> sets the angle </param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public AngleCircleLineFact(string Cid1, string Rid2, float angle, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;
        this.Rid2 = Rid2;
        this.angle = angle;

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

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

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

        // init it with 0 degrees, so we don't accidentally generate orthogonalfacts 
        // and the parsing works correctly if smb ever adds a scroll for this
        float angle = 0.0f;

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


        CircleUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).uri;
        RayUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[1]).uri;

        if (StageStatic.stage.factState.ContainsKey(CircleUri)
         && StageStatic.stage.factState.ContainsKey(RayUri))

            return new AngleCircleLineFact(CircleUri, RayUri, angle, uri, StageStatic.stage.factState);
        else return null; //If dependent facts do not exist return null
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "∠" + _Facts[Cid1].Label + _Facts[Rid2].Label;

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="val">Angle != 90f, _not checked_</param>
    /// <param name="c1URI"><see cref="Cid1"/></param>
    /// <param name="R2URI"><see cref="Rid2"/></param>
    /// <param name="val"><see cref="angle"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(float val, string c1URI, string r2URI)
    {
        MMTTerm lhs =
            new OMA(
                new OMS(MMTURIs.AnglePlaneLine),
                new List<MMTTerm> {
                    new OMS(c1URI),
                    new OMS(r2URI),
                }
            );

        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[] { Cid1, Rid2 };

    /// \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.Cid1].Label;
        obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Rid2].Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    public override int GetHashCode()
        => this.Cid1.GetHashCode() ^ this.Rid2.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(AngleCircleLineFact f1, AngleCircleLineFact f2)
    {
        if (f1.Cid1 == f2.Cid1 && f1.Rid2 == f2.Rid2)
            return true;

        CircleFact c1f1 = (CircleFact)_Facts[f1.Cid1];
        RayFact r2f1 = (RayFact)_Facts[f1.Rid2];

        CircleFact c1f2 = (CircleFact)_Facts[f2.Cid1];
        RayFact r2f2 = (RayFact)_Facts[f2.Rid2];

        return (c1f1.Equivalent(c1f2) && r2f1.Equivalent(r2f2));
    }
}

/// <summary>
/// A RadiusFact that corresponds to a <see cref="CircleFact">PointFacts</see> and has a float value (the actual radius).
/// </summary>
public class RadiusFact : FactWrappedCRTP<RadiusFact>
{
    /// \copydoc Fact.s_type
    [JsonProperty]
    protected static new string s_type = "RadiusFact";

    /// <summary> The circle corresponding to the radius </summary>
    public string Cid1;
    /// <summary> The radius as a float </summary>
    public float rad;

    /// <summary> \copydoc Fact.Fact </summary>
    public RadiusFact() : base()
    {
        this.Cid1 = null;
        this.rad = 0.0f;
    }

    /// <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 RadiusFact(RadiusFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Cid1]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public RadiusFact(string cid1, FactOrganizer organizer) : base(organizer)
        => init(cid1);

    /// <summary>
    /// Initiates <see cref="Cid1"/> and <see cref="rad"/>
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    private void init(string cid1)
    {
        this.Cid1 = cid1;

        CircleFact cf1 = _Facts[cid1] as CircleFact;
        this.rad = cf1.radius;

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, this.rad);
        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="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public RadiusFact(string Cid1, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;

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

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static RadiusFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;
        string CircleUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).uri;

        if (StageStatic.stage.factState.ContainsKey(CircleUri))

            return new RadiusFact(CircleUri, uri, StageStatic.stage.factState);

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

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "r " + _Facts[Cid1].Label;

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="rad"> see <see cref="rad"/></param>
    /// <param name="c1URI"> see <see cref="Cid1"/></param>

    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string c1URI, float rad)
    {
        MMTTerm lhs =
            new OMA(
                new OMS(MMTURIs.RadiusCircleMetric),
                new List<MMTTerm> {
                    new OMS(c1URI),
                }
            );

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

        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[] { Cid1 };

    /// \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 = "r: " + _Facts[this.Cid1].Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    public override int GetHashCode()
        => this.Cid1.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(RadiusFact f1, RadiusFact f2)
    {
        if (f1.Cid1 == f2.Cid1) // if they correspond to the same circle, then automatically the radius has to be the same
            return true;

        CircleFact c1f1 = (CircleFact)_Facts[f1.Cid1];
        CircleFact c1f2 = (CircleFact)_Facts[f2.Cid1];
        return (c1f1.Equivalent(c1f2));
    }
}

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

    /// <summary> the circle <see cref="CircleFact">CircleFact</see>  </summary>
    public string Cid1;
    /// <summary> the area which is contained by the circle </summary>
    public float A;


    /// <summary> \copydoc Fact.Fact </summary>
    public AreaCircleFact() : base()
    {
        this.Cid1 = null;
        this.A = 0.0f;
    }

    /// <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 AreaCircleFact(AreaCircleFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Cid1]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public AreaCircleFact(string cid1, FactOrganizer organizer) : base(organizer)
        => init(cid1);

    /// <summary>
    /// Initiates <see cref="Cid1"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>

    private void init(string cid1)
    {
        this.Cid1 = cid1;

        CircleFact cf1 = _Facts[cid1] as CircleFact;
        this.A = cf1.radius * cf1.radius * ((float)System.Math.PI);

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, this.A);
        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="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public AreaCircleFact(string Cid1, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;

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

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static AreaCircleFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;
        string CircleUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).uri;

        if (StageStatic.stage.factState.ContainsKey(CircleUri))
            return new AreaCircleFact(CircleUri, uri, StageStatic.stage.factState);
        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "A(" + _Facts[Cid1].Label + ")";

    /// <summary>
    /// Constructs a response, that is sent to the MMT-Server
    /// </summary>
    /// <param name="area"> area of the circle </param>
    /// <param name="c1URI">  <see cref="Cid1"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string c1URI, float area)
    {
        MMTTerm lhs =
            new OMA(
                new OMS(MMTURIs.AreaCircle),
                new List<MMTTerm> {
                    new OMS(c1URI),
                }
            );

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

        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[] { Cid1 };

    /// \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.Cid1].Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    /// is this a problem?
    public override int GetHashCode()
        => this.Cid1.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(AreaCircleFact f1, AreaCircleFact f2)
    {
        if (f1.Cid1 == f2.Cid1)
            return true;

        CircleFact c1f1 = (CircleFact)_Facts[f1.Cid1];
        CircleFact c1f2 = (CircleFact)_Facts[f2.Cid1];
        return (c1f1.Equivalent(c1f2) && f1.A == f2.A);
    }
}

/// <summary>
/// The volume of a cone A  defined by a base area  <see cref="CircleFact">CircleFact</see>, an apex <see cref="PointFact">PointFact</see> and the volume as float
/// </summary>
public class ConeVolumeFact : FactWrappedCRTP<ConeVolumeFact>
{
    /// \copydoc Fact.s_type
    [JsonProperty]
    protected static new string s_type = "ConeVolumeFact";

    ///  <summary> a <see cref="CircleFact">CircleFact</see> describing the base area </summary>
    public string Cid1;
    ///  <summary> a <see cref="PointFact">PointFact</see> describing the apex point  </summary>
    public string Pid1;
    ///  <summary> the volume of the cone as a float </summary>
    public float vol;

    /// <summary> \copydoc Fact.Fact </summary>
    public ConeVolumeFact() : base()
    {
        this.Cid1 = null;
        this.Pid1 = null;
        this.vol = 0.0f;
    }

    /// <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 ConeVolumeFact(ConeVolumeFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Cid1], old_to_new[fact.Pid1], fact.vol);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="vol">sets <see cref="vol"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public ConeVolumeFact(string cid1, string pid1, float vol, FactOrganizer organizer) : base(organizer)
        => init(cid1, pid1, vol);

    /// <summary>
    /// Initiates <see cref="Cid1"/>, <see cref="Pid1"/>, <see cref="vol"/>,  <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="vol">sets <see cref="vol"/></param>
    private void init(string cid1, string pid1, float vol)
    {
        this.Cid1 = cid1;
        this.Pid1 = pid1;
        this.vol = vol;

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, pid1, vol);
        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="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="Pid1">sets <see cref="Pid1"/></param>
    /// <param name="volume">sets <see cref="vol"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public ConeVolumeFact(string Cid1, string Pid1, float volume, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;
        this.Pid1 = Pid1;
        this.vol = volume;

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

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

        if (((Scroll.ScrollValueFact)fact).lhs == null)
            return null;

        string CircleUri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[0]).uri;
        string PointUri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[1]).uri;
        float volume = 0.0f;
        if ((((Scroll.ScrollValueFact)fact).value) != null)
            volume = ((OMF)((Scroll.ScrollValueFact)fact).value).f;

        if (StageStatic.stage.factState.ContainsKey(CircleUri) && StageStatic.stage.factState.ContainsKey(PointUri))

            return new ConeVolumeFact(CircleUri, PointUri, volume, uri, StageStatic.stage.factState);

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

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "V(" + _Facts[Cid1].Label + "," + _Facts[Pid1].Label + ")";

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="c1URI"> Uri for <see cref="Cid1"/></param>
    /// <param name="p1URI"> Uri for <see cref="Pid1"/></param>
    /// <param name="val"> <see cref="vol"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string c1URI, string p1URI, float val)
    {
        MMTTerm lhs =
            new OMA(
                new OMS(MMTURIs.VolumeCone),

                new List<MMTTerm> {
                    new OMA(new OMS(MMTURIs.ConeOfCircleApex),
                        new List<MMTTerm> {
                            new OMS(c1URI),
                            new OMS(p1URI),
                         }
                    ),
                }
            );

        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[] { Cid1, Pid1 };

    /// \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.Cid1].Label + _Facts[this.Pid1].Label;
        obj.GetComponent<FactWrapper>().fact = this;

        return obj;
    }

    /// \copydoc Fact.GetHashCode
    /// uhhh is this a problem?
    public override int GetHashCode()
        => this.Cid1.GetHashCode() ^ this.Pid1.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(ConeVolumeFact f1, ConeVolumeFact f2)
    {
        if (f1.Cid1 == f2.Cid1 && f1.Pid1 == f2.Pid1)
            return true;

        CircleFact c1f1 = (CircleFact)_Facts[f1.Cid1];
        CircleFact c1f2 = (CircleFact)_Facts[f2.Cid1];

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

        return (c1f1.Equivalent(c1f2) && p1f1.Equivalent(p1f2) && (Mathf.Abs(f1.vol - f2.vol) < 0.001));
    }
}

/// <summary>
/// The fact that the plane of a <see cref="CircleFact">CircleFact</see> and the line <see cref="RayFact>RayFact</see> are orthogonal
/// </summary>
public class OrthogonalCircleLineFact : FactWrappedCRTP<OrthogonalCircleLineFact>
{
    /// \copydoc Fact.s_type
    [JsonProperty]
    protected static new string s_type = "OrthogonalCircleLineFact";

    ///  <summary> a <see cref="CircleFact">CircleFact</see> describing the base area </summary>
    public string Cid1;
    ///  <summary> a <see cref="RayFact">Rayfact</see> describing the line </summary>
    public string Lid1;


    /// <summary> \copydoc Fact.Fact </summary>
    public OrthogonalCircleLineFact() : base()
    {
        this.Cid1 = null;
        this.Lid1 = 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 OrthogonalCircleLineFact(OrthogonalCircleLineFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Cid1], old_to_new[fact.Lid1]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="lid1">sets <see cref="Lid1"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public OrthogonalCircleLineFact(string cid1, string lid1, FactOrganizer organizer) : base(organizer)
        => init(cid1, lid1);

    /// <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="cid1">sets <see cref="Cid1"/></param>
    /// <param name="lid1">sets <see cref="Lid1"/></param>
    private void init(string cid1, string lid1)
    {
        this.Cid1 = cid1;
        this.Lid1 = lid1;

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, lid1);
        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="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="Lid1">sets <see cref="Lid1"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public OrthogonalCircleLineFact(string Cid1, string Lid1, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;
        this.Lid1 = Lid1;

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

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

        string uri = fact.@ref.uri;

        string CircleUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
        string LineUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;

        if (StageStatic.stage.factState.ContainsKey(CircleUri)
         && StageStatic.stage.factState.ContainsKey(LineUri))

            return new OrthogonalCircleLineFact(CircleUri, LineUri, uri, StageStatic.stage.factState);

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

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => _Facts[Cid1].Label + "⊥" + _Facts[Lid1].Label;

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="c1URI"> Uri for <see cref="Cid1"/></param>
    /// <param name="l1URI"> Uri for <see cref="Lid1"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string c1URI, string l1URI)
    {
        MMTTerm tp = new OMA(
            new OMS(MMTURIs.Ded), 
            new List<MMTTerm>{
                new OMA(
                    new OMS(MMTURIs.OrthoCircleLine), 
                    new List<MMTTerm>{
                        new OMS(c1URI),
                        new OMS(l1URI),
                    }
                ),
            }
        );

        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[] { Cid1, Lid1 };

    /// \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.Cid1].Label;
        obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Lid1].Label;

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

    /// \copydoc Fact.GetHashCode
    /// uhhh is this a problem?
    public override int GetHashCode()
        => this.Cid1.GetHashCode() ^ this.Lid1.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(OrthogonalCircleLineFact f1, OrthogonalCircleLineFact f2)
    {
        if (f1.Cid1 == f2.Cid1 && f1.Lid1 == f2.Lid1)
            return true;

        CircleFact c1f1 = (CircleFact)_Facts[f1.Cid1];
        CircleFact c1f2 = (CircleFact)_Facts[f2.Cid1];

        RayFact l1f1 = (RayFact)_Facts[f1.Lid1];
        RayFact l1f2 = (RayFact)_Facts[f2.Lid1];

        return (c1f1.Equivalent(c1f2) && l1f1.Equivalent(l1f2));
    }
}

/// <summary>
/// The volume of a cone A  defined by a base area  <see cref="CircleFact">CircleFact</see>, a top area <see cref="CircleFact">CircleFact</see> and the volume as float
/// </summary>
public class TruncatedConeVolumeFact : FactWrappedCRTP<TruncatedConeVolumeFact>
{
    /// \copydoc Fact.s_type
    [JsonProperty]
    protected static new string s_type = "TruncatedConeVolumeFact";

    ///  <summary> a <see cref="CircleFact">CircleFact</see> describing the base area </summary>
    public string Cid1;
    ///  <summary> a <see cref="CircleFact">CircleFact</see> describing the top area  </summary>
    public string Cid2;
    ///  <summary> the volume of Truncated the cone as a float </summary>
    public float vol;
    /// <summary> a proof that both circles have not the same size </summary>
    public string unequalCirclesProof;
    ///  <summary> OMA proof that the two circles are parallel  </summary>
    public OMA proof;


    /// <summary> \copydoc Fact.Fact </summary>
    public TruncatedConeVolumeFact() : base()
    {
        this.Cid1 = null;
        this.Cid2 = null;
        this.vol = 0.0f;
        this.unequalCirclesProof = null;
        this.proof = 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 TruncatedConeVolumeFact(TruncatedConeVolumeFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer) 
        => init(old_to_new[fact.Cid1], old_to_new[fact.Cid2], fact.vol, old_to_new[fact.unequalCirclesProof], fact.proof);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    /// <param name="vol">sets <see cref="vol"/></param>
    /// <param name="proof">sets <see cref="proof"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public TruncatedConeVolumeFact(string cid1, string cid2, float vol, string unequalproof, OMA proof, FactOrganizer organizer) : base(organizer) 
        => init(cid1, cid2, vol, unequalproof, proof);

    /// <summary>
    /// sets variables and generates MMT Declaration
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    /// <param name="vol">sets <see cref="vol"/></param>
    /// <param name="proof">sets <see cref="proof"/></param>
    private void init(string cid1, string cid2, float vol, string unequalproof, OMA proof)
    {
        this.Cid1 = cid1;
        this.Cid2 = cid2;
        this.proof = proof;
        this.unequalCirclesProof = unequalproof;
        this.vol = vol;

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, cid2, vol, unequalproof, proof);
        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="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="Cid2">sets <see cref="Cid2"/></param>
    /// <param name="volume">sets <see cref="vol"/></param>
    /// <param name="proof">sets <see cref="proof"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public TruncatedConeVolumeFact(string Cid1, string Cid2, float volume, string unequalproof, OMA proof, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;
        this.Cid2 = Cid2;