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UI_OnOff_InputModule.cs.meta

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  • Fact.cs 146.03 KiB
    using System;
    using System.Collections.Generic;
    using UnityEngine;
    using UnityEngine.Networking;
    using TMPro;
    using Newtonsoft.Json;
    using static JSONManager;
    using static CommunicationEvents;
    
    
    public class ParsingDictionary {
        //TODO? get rid of this, use reflection? instead, if possible
        //TODO: docu
        public static Dictionary<string, Func<Scroll.ScrollFact, Fact>> parseFactDictionary = new Dictionary<string, Func<Scroll.ScrollFact, Fact>>() {
            {MMTURIs.Point, PointFact.parseFact},
            {MMTURIs.Metric, LineFact.parseFact},
            {MMTURIs.Angle, AngleFact.parseFact},
            {MMTURIs.LineType, RayFact.parseFact},
            {MMTURIs.LineOf, RayFact.parseFact},
    
            {MMTURIs.OnLine, OnLineFact.parseFact},
            //90Degree-Angle
            {MMTURIs.Eq, AngleFact.parseFact},
            //Parallel-LineFact
            {MMTURIs.ParallelLine, ParallelLineFact.parseFact},
            //CircleFact
            {MMTURIs.CircleType3d, CircleFact.parseFact},
            {MMTURIs.OnCircle, OnCircleFact.parseFact },
            {MMTURIs.AnglePlaneLine, AngleCircleLineFact.parseFact },
            {MMTURIs.RadiusCircleMetric, RadiusFact.parseFact },
            {MMTURIs.AreaCircle, AreaCircleFact.parseFact },
            {MMTURIs.OrthoCircleLine, OrthogonalCircleLineFact.parseFact },
            {MMTURIs.VolumeCone ,ConeVolumeFact.parseFact  },
            {MMTURIs.TruncatedVolumeCone ,TruncatedConeVolumeFact.parseFact  },
            {MMTURIs.RightAngle, RightAngleFact.parseFact },
            {MMTURIs.CylinderVolume, CylinderVolumeFact.parseFact },
            {MMTURIs.TestType, TestFact.parseFact },
            {MMTURIs.EqualityCircles, EqualCirclesFact.parseFact },
            {MMTURIs.UnEqualityCircles, UnEqualCirclesFact.parseFact }
    
    
    
    
        };
        /// Current solution to retrieve the fact ID from 
        
        public static string MMTermToString (MMTTerm term){
            if(term == null)
                return null;
            // case for OMA 
            if( term is OMA){
                OMA term_casted = (OMA) term;
                string applicant = ((OMS)term_casted.applicant).uri;
                string argument = "";
                for (int i = 0; i < term_casted.arguments.Count; i++) {
                    argument = argument+ " " + MMTermToString(term_casted.arguments[i]);
                }
                return " " + applicant + " "+ argument;
    
            }
            // case for OMS 
            if (term is OMS)
            {
                OMS term_casted = (OMS)term;  
                return term_casted.uri;
    
            }
    
            // case for OMF
            if (term is OMF) {
                OMF term_casted = (OMF)term;
                return term_casted.f.ToString();
    
            }
            
    
    
            return "couldn't understand the type";
    
    
        }
    
        public static Dictionary<string, string> parseTermsToId = new Dictionary<string, string>();
    
    
    }
    
    /// <summary>
    /// class to Read AddFact Responses.
    /// </summary>
    // TODO: docu
    public class AddFactResponse
    {
        public string uri;
    
        public static bool sendAdd(MMTDeclaration mmtDecl, out string uri)
        {
            string body = MMTSymbolDeclaration.ToJson(mmtDecl);
            return sendAdd(CommunicationEvents.ServerAdress + "/fact/add", body, out uri);
        }
    
        public static bool sendAdd(string path, string body, out string uri)
        {
            if (!CommunicationEvents.ServerRunning)
            {
                Debug.LogWarning("Server not running");
                uri = null;
                return false;
            }
    
            if(VerboseURI)
                Debug.Log("Sending to Server:\n" + body);
    
            //Put constructor parses stringbody to byteArray internally  (goofy workaround)
            UnityWebRequest www = UnityWebRequest.Put(path, body);
            www.method = UnityWebRequest.kHttpVerbPOST;
            www.SetRequestHeader("Content-Type", "application/json");
            www.timeout = 1;
    
            //TODO: implement real asynchronous communication ...
            AsyncOperation op = www.SendWebRequest();
            while (!op.isDone) ;
    
            if (www.result == UnityWebRequest.Result.ConnectionError
             || www.result == UnityWebRequest.Result.ProtocolError)
            {
                Debug.LogWarning(www.error);
                uri = null;
                return false;
            }
            else
            {
                string answer = www.downloadHandler.text;
                AddFactResponse res = JsonUtility.FromJson<AddFactResponse>(answer);
    
                if (VerboseURI)
                    Debug.Log("Server added Fact:\n" + res.uri);
    
                uri = res.uri;
                return true;
            }
        }
    }
    
    /// <summary>
    /// %Fact representation of Unity; mostly mirrors Facts of MMT.
    /// </summary>
    public abstract class Fact
    {
        /// <summary>
        /// Reference to <c>GameObject</c> that represents this Fact in the GameWorld.
        /// </summary>
        /// <seealso cref="FactObject"/>
        [JsonIgnore]
        public GameObject Representation;
    
        /// <value>
        /// Unique Id. e.g.: MMT URI
        /// </value>
        public string Id { 
            get { return _URI; }
            set { if (_URI == null) _URI = value; } // needed for JSON
        }
    
        /// <summary>
        /// MMT URI
        /// </summary>
        protected string _URI;
    
        /// <value>
        /// <c>get</c> initiates and subsequently updates a human readable name. <remarks>Should be called once a constructor call to be initiated.</remarks>
        /// <c>set</c> calls <see cref="rename(string)"/>
        /// </value>
        public string Label {
            get { // in case of renamed dependables
                return hasCustomLabel && _CustomLabel != null ?
                    _CustomLabel :
                    generateLabel();
            }
            set { rename(value); }
        }
    
        /// <value>
        /// Is true if Fact has a custom <see cref="Label"/> which is not <c>null</c> or <c>""</c>.
        /// </value>
        public bool hasCustomLabel {
            get { return LabelId < 0; }
        }
        /// <summary>
        /// Stores custom <see cref="Label"/> if set.
        /// </summary>
        protected string _CustomLabel = null;
    
        /// <summary>
        /// Counter to organize auto generated <see cref="Label"/>.
        /// Set to negative, if custom \ref Label is assigned.
        /// </summary>
        // property for JSON to set AFTER Label => declare AFTER Label
        public int LabelId { get; set; }
    
        /// <summary>
        /// Reference to <see cref="FactOrganizer"/> in which this Fact and all its <see cref="getDependentFactIds">depending Facts</see> are beeing organized.
        /// </summary>
        protected FactOrganizer _Facts;
    
        /// <summary>
        /// Only being used by [JsonReader](https://www.newtonsoft.com/json/help/html/T_Newtonsoft_Json_JsonReader.htm) to initiate empty \ref Fact "Facts".
        /// <seealso cref="JSONManager"/>
        /// </summary>
        protected Fact()
        {
            this._Facts = new FactOrganizer();
            LabelId = 0;
        }
    
        /// <summary>
        /// Standard base-constructor.
        /// </summary>
        /// <param name="organizer"><see cref="_Facts"/></param>
        protected Fact(FactOrganizer organizer)
        {
            this._Facts = organizer;
            LabelId = 0;
        }
    
        /// <summary>
        /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
        /// </summary>
        /// <param name="fact">Fact to be copied</param>
        /// <param name="organizer"><see cref="_Facts"/></param>
        protected Fact(Fact fact, FactOrganizer organizer)
        {
            this._Facts = organizer;
            LabelId = fact.LabelId;
    
            if (hasCustomLabel)
                _CustomLabel = fact.Label;
        }
    
        /// <summary>
        /// Assignes a custom <see cref="Label"/>, if <paramref name="newLabel"/> is not yet taken;
        /// or clears custom <see cref="Label"/>.
        /// </summary>
        /// <param name="newLabel">To be new <see cref="Label"/>. To reset to auto-generation set to <c>null</c> or <c>""</c>.</param>
        /// <returns></returns>
    
        //TODO: notify about updated dependable Labelnames for UI
        //TODO: check for colissions with not yet generated names
        public bool rename(string newLabel)
        // returns true if succeded
        {
            if (string.IsNullOrEmpty(newLabel))
            // switch back to autogenerated
            {
                generateLabel();
                _CustomLabel = null;
                return true;
            }
            else
            // set CustomLabel if available
            {
                if (_Facts.ContainsLabel(newLabel))
                    return false;
    
                freeAutoLabel();
                _CustomLabel = newLabel;
    
                return true;
            }
        }
    
        /// <returns><see langword="true"/> if Fact depends on other \ref Fact "Facts"; equivalent to <see cref="getDependentFactIds"/> returns non empty array</returns>
        public abstract bool hasDependentFacts();
    
        /// <returns> array of Fact <see cref="Id"> Ids </see> on which this Fact depends.</returns>
        /// <example><see cref="AngleFact"/> needs 3 <see cref="PointFact"/>s to be defined.</example>
        public abstract string[] getDependentFactIds();
    
        /// <summary>
        /// Initiates a <paramref name="prefab"/> at <paramref name="transform"/> e.g. by setting <see cref="Label"/>.
        /// </summary>
        /// <remarks>Does not set <see cref="Representation"/>.</remarks>
        /// <param name="prefab"><c>GameObject</c> Prefab that will represent this Fact</param>
        /// <param name="transform"><c>Transform</c> where to initiate <paramref name="prefab"/></param>
        /// <returns></returns>
    
        // TODO: set Representation here instead of ...
        public abstract GameObject instantiateDisplay(GameObject prefab, Transform transform);
    
        /// <summary>
        /// Frees ressources e.g. <see cref="Label"/> and will eventually delete %Fact Server-Side in far-near future when feature is supported.
        /// </summary>
        /// <param name="keep_clean">when set to <c>true</c> will upkeep <see cref="Label"/> organization.</param>
    
        // TODO? replace by ~Fact() { }
        public virtual void delete(bool keep_clean = true)
        {
            //TODO: MMT: delete over there
    
            if (keep_clean)
                freeAutoLabel();
    
            if (VerboseURI)
                Debug.Log("Server removed Fact:\n" + this.Id);
        }
    
        /// <summary>
        /// Compares \ref Fact "this" against <paramref name="f2"/>.
        /// </summary>
        /// <param name="f2">Fact to compare to</param>
        /// <returns><c>true</c> if <paramref name="f2"/> is semantical very similar to \ref Fact "this"</returns>
        public abstract bool Equivalent(Fact f2);
    
        /// <summary>
        /// Compares <paramref name="f1"/> against <paramref name="f2"/>.
        /// </summary>
        /// <param name="f1">Fact to compare to</param>
        /// <param name="f2">Fact to compare to</param>
        /// <returns><c>true</c> if <paramref name="f2"/> is semantical very similar to <paramref name="f1"/></returns>
        public abstract bool Equivalent(Fact f1, Fact f2);
    
        /// <summary>
        /// canonical
        /// </summary>
        /// <returns>unique-ish Hash</returns>
        public abstract override int GetHashCode();
    
        /// <summary>
        /// auto-generates <see cref="Label"/> using generation variable(s) e.g. <see cref="LabelId"/>;
        /// if custom <see cref="Label"/> is set, tries to restore original generated <see cref="Label"/> **without** resetting <see cref="_CustomLabel"/>. If original <see cref="Label"/> is already taken, a new one will be generated.
        /// </summary>
        /// <returns>auto-generated <see cref="Label"/></returns>
        protected virtual string generateLabel()
        {
            if (LabelId < 0)
            // reload Label if possible
                LabelId = _Facts.UnusedLabelIds.Remove(-LabelId) ? -LabelId : 0;
    
            if (LabelId == 0)
                if (_Facts.UnusedLabelIds.Count == 0)
                    LabelId = ++_Facts.MaxLabelId;
                else
                {
                    LabelId = _Facts.UnusedLabelIds.Min;
                    _Facts.UnusedLabelIds.Remove(LabelId);
                }
    
            return ((char)(64 + LabelId)).ToString();
        }
    
        /// <summary>
        /// Parses <see cref="Scroll.ScrollFact"/> to actual Fact
        /// </summary>
        /// <param name="fact">instance to be parsed</param>
        /// <returns>parsed Fact</returns>
        public static Fact parseFact(Scroll.ScrollFact fact)
        {
            return null;
        }
    
        /// <summary>
        /// Tells <see cref="_Facts"/> that \ref Fact "this" no longer uses auto-generated <see cref="Label"/>, but remembers current generation variable(s).
        /// </summary>
    
        // TODO? only get _Fact to freeLabel/
        public /*protected internal*/ void freeAutoLabel()
        {
            if (LabelId > 0)
            {
                _Facts.UnusedLabelIds.Add(LabelId);
                // store Label for name-persistance
                LabelId = -LabelId;
            }
        }
    }
    
    /// <summary>
    /// Implements CRTP for <see cref="Fact"/>; Escalates constructors;
    /// </summary>
    /// <typeparam name="T">class, which inherits from FactWrappedCRTP</typeparam>
    public abstract class FactWrappedCRTP<T>: Fact where T: FactWrappedCRTP<T>
    {
        /// <summary>\copydoc Fact.Fact()</summary>
        protected FactWrappedCRTP() : base() { }
    
        /// <summary>\copydoc Fact.Fact(FactOrganizer)</summary>
        protected FactWrappedCRTP(FactOrganizer organizer) : base(organizer) { }
    
        /// <summary>\copydoc Fact.Fact(Fact, FactOrganizer)</summary>
        protected FactWrappedCRTP(FactWrappedCRTP<T> fact, FactOrganizer organizer) : base(fact, organizer) { }
    
        /// \copydoc Fact.Equivalent(Fact)
        public override bool Equivalent(Fact f2)
        {
            return Equivalent(this, f2);
        }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        public override bool Equivalent(Fact f1, Fact f2)
        {
            return f1.GetType() == f2.GetType() && EquivalentWrapped((T)f1, (T)f2);
        }
    
        /// <summary>CRTP step of <see cref="Equivalent(Fact)"/> and <see cref="Equivalent(Fact, Fact)"/></summary>
        protected abstract bool EquivalentWrapped(T f1, T f2);
    }
    
    /// <summary>
    /// Base-class for 1D-Facts
    /// </summary>
    public abstract class AbstractLineFact: FactWrappedCRTP<AbstractLineFact>
    {
        /// @{ <summary>
        /// One <see cref="Fact.Id">Id</see> of two <see cref="PointFact"/> defining <see cref="Dir"/>.
        /// </summary>
        public string Pid1, Pid2;
        /// @}
    
        /// <summary>
        /// Normalized Direction from <see cref="Pid1"/> to <see cref="Pid2"/>.
        /// </summary>
        public Vector3 Dir;
    
        /// <summary>
        /// \copydoc Fact.Fact()
        /// </summary>
        protected AbstractLineFact() : base()
        {
            Pid1 = null;
            Pid2 = null;
            Dir = Vector3.zero;
        }
    
        /// <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>
        protected AbstractLineFact(AbstractLineFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        {
            set_public_members(old_to_new[fact.Pid1], old_to_new[fact.Pid2]);
        }
    
        /// <summary>
        /// Standard Constructor
        /// </summary>
        /// <param name="pid1">sets <see cref="AbstractLineFact.Pid1"/></param>
        /// <param name="pid2">sets <see cref="AbstractLineFact.Pid2"/></param>
        /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
        protected AbstractLineFact(string pid1, string pid2, FactOrganizer organizer): base(organizer)
        {
            set_public_members(pid1, pid2);
        }
    
        /// <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="backendURI">MMT URI</param>
        /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
        protected AbstractLineFact(string pid1, string pid2, string backendURI, FactOrganizer organizer) : base(organizer)
        {
            set_public_members(pid1, pid2);
            this._URI = backendURI;
        }
    
        /// <summary>
        /// Initiates <see cref="Pid1"/>, <see cref="Pid2"/>, <see cref="Dir"/>
        /// </summary>
        /// <param name="pid1">sets <see cref="Pid1"/></param>
        /// <param name="pid2">sets <see cref="Pid2"/></param>
        private void set_public_members(string pid1, string pid2)
        {
            this.Pid1 = pid1;
            this.Pid2 = pid2;
            PointFact pf1 = _Facts[pid1] as PointFact;
            PointFact pf2 = _Facts[pid2] as PointFact;
            this.Dir = (pf2.Point - pf1.Point).normalized;
        }
    
        /// \copydoc Fact.hasDependentFacts
        public override bool hasDependentFacts()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return new string[] { Pid1, Pid2 };
        }
    
        /// \copydoc Fact.GetHashCode
        public override int GetHashCode()
        {
            return this.Pid1.GetHashCode() ^ this.Pid2.GetHashCode();
        }
    }
    
    /// <summary>
    /// Implements CRTP for <see cref="AbstractLineFact"/>; Escalates constructors;
    /// </summary>
    /// <typeparam name="T">class, which inherits from AbstractLineFactWrappedCRTP</typeparam>
    public abstract class AbstractLineFactWrappedCRTP<T>: AbstractLineFact where T: AbstractLineFactWrappedCRTP<T>
    {
        /// <summary>\copydoc Fact.Fact</summary>
        protected AbstractLineFactWrappedCRTP () : base() { }
    
        /// <summary>\copydoc AbstractLineFact.AbstractLineFact(AbstractLineFact, Dictionary{string, string}, FactOrganizer)</summary>
        protected AbstractLineFactWrappedCRTP (AbstractLineFactWrappedCRTP<T> fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, old_to_new, organizer) { }
    
        /// <summary>\copydoc AbstractLineFact.AbstractLineFact(string, string, FactOrganizer)</summary>
        protected AbstractLineFactWrappedCRTP (string pid1, string pid2, FactOrganizer organizer) : base(pid1, pid2, organizer) { }
    
        /// <summary>\copydoc AbstractLineFact.AbstractLineFact(string, string, string, FactOrganizer)</summary>
        protected AbstractLineFactWrappedCRTP (string pid1, string pid2, string backendURI, FactOrganizer organizer) : base(pid1, pid2, backendURI, organizer) { }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        protected override bool EquivalentWrapped(AbstractLineFact f1, AbstractLineFact f2)
        {
            return EquivalentWrapped((T)f1, (T)f2);
        }
    
        /// <summary>CRTP step of <see cref="EquivalentWrapped(AbstractLineFact, AbstractLineFact)"/></summary>
        protected abstract bool EquivalentWrapped(T f1, T f2);
    }
    
    /// <summary>
    /// Point in 3D Space
    /// </summary>
    public class PointFact : FactWrappedCRTP<PointFact>
    {
        /// <summary> Position </summary>
        public Vector3 Point;
        /// <summary> Orientation for <see cref="Fact.Representation"/> </summary>
        public Vector3 Normal;
    
    
        /// <summary> \copydoc Fact.Fact </summary>
        public PointFact() : base()
        {
            this.Point = Vector3.zero;
            this.Normal = Vector3.zero;
        }
    
        /// <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 PointFact(PointFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        {
            init(fact.Point, fact.Normal);
        }
    
        /// <summary>
        /// Standard Constructor
        /// </summary>
        /// <param name="P">sets <see cref="Point"/></param>
        /// <param name="N">sets <see cref="Normal"/></param>
        /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
        public PointFact(Vector3 P, Vector3 N, FactOrganizer organizer) : base(organizer)
        {
            init(P, N);
        }
    
        /// <summary>
        /// Initiates <see cref="Point"/>, <see cref="Normal"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
        /// </summary>
        /// <param name="P">sets <see cref="Point"/></param>
        /// <param name="N">sets <see cref="Normal"/></param>
        private void init(Vector3 P, Vector3 N)
        {
            this.Point = P;
            this.Normal = N;
    
            List<MMTTerm> arguments = new List<MMTTerm>
            {
                new OMF(P.x),
                new OMF(P.y),
                new OMF(P.z)
            };
    
            //OMS constructor generates full URI
            MMTTerm tp = new OMS(MMTURIs.Point);
            MMTTerm df = new OMA(new OMS(MMTURIs.Tuple), arguments);
    
    
            MMTSymbolDeclaration mmtDecl = new MMTSymbolDeclaration(this.Label, tp, df);
            AddFactResponse.sendAdd(mmtDecl, out this._URI);
            string parse_id = ParsingDictionary.MMTermToString(df);
            ParsingDictionary.parseTermsToId[parse_id] = this._URI;
    
    
        }
    
        /// <summary>
        /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
        /// <see cref="Normal"/> set to <c>Vector3.up</c>
        /// </summary>
        /// <param name="a">sets <c>x</c> coordinate of <see cref="Point"/></param>
        /// <param name="b">sets <c>y</c> coordinate of <see cref="Point"/></param>
        /// <param name="c">sets <c>z</c> coordinate of <see cref="Point"/></param>
        /// <param name="uri">MMT URI</param>
        /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
        public PointFact(float a, float b, float c, string uri, FactOrganizer organizer) : base(organizer)
        {
            this.Point = new Vector3(a, b, c);
            this.Normal = Vector3.up;
            this._URI = uri;
            _ = this.Label;
        }
    
        /// \copydoc Fact.parseFact(Scroll.ScrollFact)
        public new static PointFact parseFact(Scroll.ScrollFact fact) {
            String uri = fact.@ref.uri;
            OMA df = (OMA)((Scroll.ScrollSymbolFact)fact).df;
    
            Debug.Log("Point decomposed :"+ParsingDictionary.MMTermToString(df));
    
    
            if (df == null)
                return null;
    
           
            float a = (float)((OMF)df.arguments[0]).f;
            float b = (float)((OMF)df.arguments[1]).f;
            float c = (float)((OMF)df.arguments[2]).f;
    
            string parse_id = ParsingDictionary.MMTermToString(df);
         
            Debug.Log("point added: " + parse_id);
            if(!ParsingDictionary.parseTermsToId.ContainsKey(parse_id))
                    ParsingDictionary.parseTermsToId[parse_id] = uri;
    
    
             return new PointFact(a, b, c, uri, StageStatic.stage.factState);
            
        }
    
        /// \copydoc Fact.hasDependentFacts
        public override Boolean hasDependentFacts() {
            return false;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds() {
            return new string[] { };
        }
    
        /// \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 = this.Label;
            obj.GetComponent<FactWrapper>().fact = this;
            return obj;
        }
    
        /// \copydoc Fact.GetHashCode
        public override int GetHashCode()
        {
            return this.Point.GetHashCode() ^ this.Normal.GetHashCode();
        }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        protected override bool EquivalentWrapped(PointFact f1, PointFact f2)
        {
            return Math3d.IsApproximatelyEqual(f1.Point, f2.Point);
        }
    
    
    }
    
    /// <summary>
    /// Line within 3D Space of finite length
    /// </summary>
    public class LineFact : AbstractLineFactWrappedCRTP<LineFact>
    {
        /// <summary> Distance between <see cref="AbstractLineFact.Pid1"/> and <see cref="AbstractLineFact.Pid2"/></summary>
        public float Distance;
    
        /// <summary> \copydoc Fact.Fact </summary>
        public LineFact() : base()
        {
            Distance = 0;
        }
    
        /// <summary> \copydoc AbstractLineFact.AbstractLineFact(AbstractLineFact, Dictionary<string, string>, FactOrganizer) </summary>
        public LineFact(LineFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, old_to_new, organizer)
        {
            init(old_to_new[fact.Pid1], old_to_new[fact.Pid2]);
        }
    
        /// <summary> \copydoc AbstractLineFact.AbstractLineFact(string, string, string, FactOrganizer) </summary>
        public LineFact(string pid1, string pid2, string backendURI, FactOrganizer organizer) : base(pid1, pid2, backendURI, organizer)
        {
            SetDistance();
            _ = this.Label;
        }
    
        /// <summary> \copydoc AbstractLineFact.AbstractLineFact(string, string, FactOrganizer) </summary>
        public LineFact(string pid1, string pid2, FactOrganizer organizer) : base(pid1, pid2, organizer)
        {
            init(pid1, pid2);
        }
    
        /// <summary>
        /// Initiates <see cref="AbstractLineFact.Pid1"/>, <see cref="AbstractLineFact.Pid2"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
        /// </summary>
        /// <param name="pid1">sets <see cref="AbstractLineFact.Pid1"/></param>
        /// <param name="pid2">sets <see cref="AbstractLineFact.Pid2"/></param>
        private void init(string pid1, string pid2)
        {
            SetDistance();
    
            PointFact pf1 = _Facts[pid1] as PointFact;
            PointFact pf2 = _Facts[pid2] as PointFact;
    
            string p1URI = pf1.Id;
            string p2URI = pf2.Id;
            float v = (pf1.Point - pf2.Point).magnitude;
    
            MMTTerm lhs =
                new OMA(
                    new OMS(MMTURIs.Metric),
                    new List<MMTTerm> {
                        new OMS(p1URI),
                        new OMS(p2URI)
                    }
                );
    
            MMTTerm valueTp = new OMS(MMTURIs.RealLit);
            MMTTerm value = new OMF(v);
    
            //see point label
            MMTValueDeclaration mmtDecl = new MMTValueDeclaration(this.Label, lhs, valueTp, value);
            AddFactResponse.sendAdd(mmtDecl, out this._URI);
        }
    
        /// \copydoc Fact.parseFact(Scroll.ScrollFact)
        public new static LineFact parseFact(Scroll.ScrollFact fact)
        {
            string uri = fact.@ref.uri;
            string pointAUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).uri;
            string pointBUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[1]).uri;
    
            if (StageStatic.stage.factState.ContainsKey(pointAUri)
             && StageStatic.stage.factState.ContainsKey(pointBUri))
                return new LineFact(pointAUri, pointBUri, uri, StageStatic.stage.factState);
    
            //If dependent facts do not exist return null
            else {
                return null;
            }
        }
    
        /// \copydoc Fact.generateLabel
        protected override string generateLabel()
        {
            return "[" + _Facts[Pid1].Label + _Facts[Pid2].Label + "]";
        }
    
        /// \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.GetComponent<FactWrapper>().fact = this;
            return obj;
        }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        protected override bool EquivalentWrapped(LineFact f1, LineFact f2)
        {
            if ((f1.Pid1 == f2.Pid1 && f1.Pid2 == f2.Pid2))// || 
                //(f1.Pid1 == f2.Pid2 && f1.Pid2 == f2.Pid1))
                return true;
    
            PointFact p1f1 = (PointFact)_Facts[f1.Pid1];
            PointFact p2f1 = (PointFact)_Facts[f1.Pid2];
            PointFact p1f2 = (PointFact)_Facts[f2.Pid1];
            PointFact p2f2 = (PointFact)_Facts[f2.Pid2];
    
            return (p1f1.Equivalent(p1f2) && p2f1.Equivalent(p2f2))
                ;//|| (p1f1.Equivalent(p2f2) && p2f1.Equivalent(p1f2));
        }
    
        /// <summary> Calculates and sets <see cref="Distance"/>; <remarks> <see cref="AbstractLineFact.Pid1"/> and <see cref="AbstractLineFact.Pid2"/> needs to be set first.</remarks></summary>
        private void SetDistance()
        {
            this.Distance = Vector3.Distance(((PointFact)_Facts[Pid1]).Point, ((PointFact)_Facts[Pid2]).Point);
        }
    }
    
    /// <summary>
    /// Ray within 3D Space of infinite length
    /// </summary>
    public class RayFact : AbstractLineFactWrappedCRTP<RayFact>
    {
        /// <summary> \copydoc Fact.Fact </summary>
        public RayFact() : base() { }
    
        /// <summary> \copydoc AbstractLineFact.AbstractLineFact(AbstractLineFact, Dictionary<string, string>, FactOrganizer) </summary>
        public RayFact(RayFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, old_to_new, organizer)
        {
            init(old_to_new[fact.Pid1], old_to_new[fact.Pid2]);
        }
    
        /// <summary> \copydoc AbstractLineFact.AbstractLineFact(string, string, string, FactOrganizer) </summary>
        public RayFact(string pid1, string pid2, string backendURI, FactOrganizer organizer) : base(pid1, pid2, backendURI, organizer)
        {
            _ = this.Label;
        }
    
        /// <summary> \copydoc AbstractLineFact.AbstractLineFact(string, string, FactOrganizer) </summary>
        public RayFact(string pid1, string pid2, FactOrganizer organizer) : base(pid1, pid2, organizer)
        {
            init(pid1, pid2);
        }
    
        /// <summary>
        /// Initiates <see cref="AbstractLineFact.Pid1"/>, <see cref="AbstractLineFact.Pid2"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
        /// </summary>
        /// <param name="pid1">sets <see cref="AbstractLineFact.Pid1"/></param>
        /// <param name="pid2">sets <see cref="AbstractLineFact.Pid2"/></param>
        private void init(string pid1, string pid2)
        {
            PointFact pf1 = _Facts[pid1] as PointFact;
            PointFact pf2 = _Facts[pid2] as PointFact;
    
            string p1URI = pf1.Id;
            string p2URI = pf2.Id;
    
            List<MMTTerm> arguments = new List<MMTTerm>
            {
                new OMS(p1URI),
                new OMS(p2URI)
            };
    
            //OMS constructor generates full URI
            MMTTerm tp = new OMS(MMTURIs.LineType);
            MMTTerm df = new OMA(new OMS(MMTURIs.LineOf), arguments);
    
            MMTSymbolDeclaration mmtDecl = new MMTSymbolDeclaration(this.Label, tp, df);
            AddFactResponse.sendAdd(mmtDecl, out this._URI);
            string parse_id = ParsingDictionary.MMTermToString(df);
            Debug.Log("point added: " + parse_id);
            ParsingDictionary.parseTermsToId[parse_id] = this._URI;
    
    
        }
    
        /// \copydoc Fact.parseFact(Scroll.ScrollFact)
        public new static RayFact parseFact(Scroll.ScrollFact fact)
        {
            string uri = fact.@ref.uri;
    
    
            // temporary fix to the problem, that actually calculating stuff leads to losing the IDS
            /**  if (fact.kind == "veq") {
                  Debug.Log("RayFact veq activated");
                  string pointAUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).uri;
                  string pointBUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[1]).uri;
                  if (StageStatic.stage.factState.ContainsKey(pointAUri)
                   && StageStatic.stage.factState.ContainsKey(pointBUri))
                      return new RayFact(pointAUri, pointBUri, uri, StageStatic.stage.factState);
    
                  return null;
              }*/
            if ((OMA)((Scroll.ScrollSymbolFact)fact).df == null)
                return null;
           
            string pointAUri = ((OMS)((OMA)((Scroll.ScrollSymbolFact)fact).df).arguments[0]).uri;
            string pointBUri = ((OMS)((OMA)((Scroll.ScrollSymbolFact)fact).df).arguments[1]).uri;
    
             if (StageStatic.stage.factState.ContainsKey(pointAUri)
                 && StageStatic.stage.factState.ContainsKey(pointBUri))
                 return new RayFact(pointAUri, pointBUri, uri, StageStatic.stage.factState);
    
           //If dependent facts do not exist return null
            
            return null;
        }
    
        /// \copydoc Fact.generateLabel
        protected override string generateLabel()
        {
           // return "–" + _Facts[Pid1].Label + _Facts[Pid2].Label + "–";
            return _Facts[Pid1].Label + _Facts[Pid2].Label ;
    
        }
    
        /// \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 = this.Label;
            obj.GetComponent<FactWrapper>().fact = this;
            return obj;
        }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        protected override bool EquivalentWrapped(RayFact f1, RayFact f2)
        {
            if (!Math3d.IsApproximatelyParallel(f1.Dir, f2.Dir))
                return false;
    
            PointFact p1f1 = (PointFact)_Facts[f1.Pid1];
            PointFact p1f2 = (PointFact)_Facts[f2.Pid1];
            PointFact p2f2 = (PointFact)_Facts[f2.Pid2];
    
            return Math3d.IsPointApproximatelyOnLine(p1f1.Point, f1.Dir, p1f2.Point)
                && Math3d.IsPointApproximatelyOnLine(p1f1.Point, f1.Dir, p2f2.Point);
        }
    }
    
    /// <summary>
    /// A <see cref="PointFact"/> on a <see cref="AbstractLineFact"/>
    /// </summary>
    public class OnLineFact : FactWrappedCRTP<OnLineFact>
    {
        public string
            /// <summary> <see cref="PointFact"/>.<see cref="Fact.Id">Id</see> </summary>
            Pid,
            /// <summary> <see cref="AbstractLineFact"/>.<see cref="Fact.Id">Id</see> </summary>
            Rid;
    
        /// <summary> \copydoc Fact.Fact </summary>
        public OnLineFact() : base()
        {
            this.Pid = null;
            this.Rid = 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 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;
    
            PointFact pf = _Facts[pid] as PointFact;
            RayFact rf = _Facts[rid] as RayFact;
            string pURI = pf.Id;
            string rURI = rf.Id;
    
            List<MMTTerm> innerArguments = new List<MMTTerm>
            {
                new OMS(rURI),
                new OMS(pURI)
            };
    
            List<MMTTerm> outerArguments = new List<MMTTerm>
            {
                new OMA(new OMS(MMTURIs.OnLine), innerArguments)
            };
    
            //OMS constructor generates full URI
            MMTTerm tp = new OMA(new OMS(MMTURIs.Ded), outerArguments);
            MMTTerm df = null;
    
            MMTSymbolDeclaration mmtDecl = new MMTSymbolDeclaration(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 lineUri = "";
            string pointUri = "";
            
            if (((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0] is OMS)
            {
                // standard case
                lineUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
                pointUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;
            }
            else {
                // case when line Uri has a projl on the line Argument 
                lineUri = ((OMS)((OMA)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).arguments[0]).uri;
                pointUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).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()
        {
            return _Facts[Pid].Label + "∈" + _Facts[Rid].Label;
        }
    
        /// \copydoc Fact.hasDependentFacts
        public override Boolean hasDependentFacts()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return 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>
    {
        /// @{ <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;
    
            PointFact pf1 = _Facts[pid1] as PointFact;
            PointFact pf2 = _Facts[pid2] as PointFact;
            PointFact pf3 = _Facts[pid3] as PointFact;
    
            float v = GetAngle(); // sets is_right_angle
            if (Mathf.Abs(Mathf.Abs(v) - 90.0f) < 0.1)
            {
                // also generate a RightAngleFact 
                RightAngleFact rightAngle = new RightAngleFact(pid1, pid2, pid3, StageStatic.stage.factState);
                // I have no clue if this is actually necessary
                bool exists;
                StageStatic.stage.factState.Add(rightAngle, out exists, true);
            }
    
    
            MMTDeclaration mmtDecl;
            string p1URI = pf1.Id;
            string p2URI = pf2.Id;
            string p3URI = pf3.Id;
           // if (is_right_angle)
           //     mmtDecl = generate90DegreeAngleDeclaration(v, p1URI, p2URI, p3URI);
           // else
              mmtDecl = generateNot90DegreeAngleDeclaration(v, p1URI, p2URI, p3URI);
    
            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;
    
            float v = GetAngle();
    
            if (Mathf.Abs(Mathf.Abs(v) - 90.0f) < 0.1)
            {
                // also generate an orthogonal circle line fact
    
                RightAngleFact rightAngle = new RightAngleFact(Pid1, Pid2, Pid3, StageStatic.stage.factState);
                // is it necessary to make a boolean variable here?
                bool exists;
                StageStatic.stage.factState.Add(rightAngle, out exists, true);
            }
    
    
    
            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;
    
            if (Mathf.Abs(Mathf.Abs(angle) - 90.0f) < 0.1)
            {
                // also generate an orthogonal circle line fact
    
                RightAngleFact rightAngle = new RightAngleFact(Pid1, Pid2, Pid3, StageStatic.stage.factState);
                // I have no clue if this is actually necessary
                bool exists;
                StageStatic.stage.factState.Add(rightAngle, out exists, true);
            }
            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.Log("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, uri, StageStatic.stage.factState);
                    return new AngleFact(pointAUri, pointBUri, pointCUri,angle, uri, StageStatic.stage.factState);
            }
            else
            {   //If dependent facts do not exist return null
                return null;
            }
        }
    
        /// \copydoc Fact.generateLabel
        protected override string generateLabel()
        {
            return (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()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return 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));
        }
    }
    
    
    /// TODO Work in Progress
    /// <summary>
    /// Two parallel Lines comprised of two <see cref="LineFact">LineFacts</see> 
    /// </summary>
    public class ParallelLineFact : FactWrappedCRTP<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)
        {
            //TODO must be parallel lines 
            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 lineAUri = "";
            string lineBUri = "";
    
            string uri = fact.@ref.uri;
            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)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
                lineBUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;
    
            }
            // Second case might be redundant by now
            else {
                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()
        {
            return "||" + _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)
        {
            
            List<MMTTerm> innerArguments = new List<MMTTerm>
            {
                new OMS(l1URI),
                new OMS(l2URI)
            };
    
            List<MMTTerm> outerArguments = new List<MMTTerm>
            {
                new OMA(new OMS(MMTURIs.ParallelLine), innerArguments)
            };
    
            //OMS constructor generates full URI
            MMTTerm tp = new OMA(new OMS(MMTURIs.Ded), outerArguments);
            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()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return 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)) ;
        }
    }
    
    
    
    
    //TODO big work in progress Circle Wrapper
    
    /// TODO Work in Progress
    /// <summary>
    /// A Circle that is made out of a middle point, a plane and a radius  
    /// </summary>
    public class CircleFact : FactWrappedCRTP<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;
    
            PointFact pf1 = _Facts[pid1] as PointFact;
            PointFact pf2 = _Facts[pid2] as PointFact;
    
    
            this.radius = radius;
            this.normal = normal;
    
            MMTDeclaration mmtDecl;
            string p1URI = pf1.Id;
            string p2URI = pf2.Id;
    
            mmtDecl = generateCircleFactDeclaration(p1URI, p2URI, 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;
            string M_uri = "";
            string A_uri = "";
            float radius = 0.0f;
            Vector3 normal = Vector3.zero;
            OMA df = (OMA)((Scroll.ScrollSymbolFact)fact).df;
    
            if (df == null)
                return null;
    
            Scroll.ScrollSymbolFact casted_fact = (Scroll.ScrollSymbolFact)fact;
            
    
    
            // get the mid point uri
            string parse_id_M = ParsingDictionary.MMTermToString(((OMA)casted_fact.df).arguments[1]);
    
            Debug.Log("parse_id_M " + parse_id_M);
            M_uri = ParsingDictionary.parseTermsToId[parse_id_M];
            Debug.Log("M URI " + M_uri);
    
            // get the radius
            radius = ((OMF)((OMA)casted_fact.df).arguments[2]).f;
    
            OMA planeOMA = (OMA)((OMA)casted_fact.df).arguments[0];
            string planeApplicant = ((OMS)planeOMA.applicant).uri;
    
    
            // Getting the plane
            // IN case of a normale plane
            if (planeApplicant.Equals(MMTURIs.pointNormalPlane))
            {
                Debug.Log("planeApplicant" + planeApplicant);
    
                OMA pointAOMA = (OMA)planeOMA.arguments[0];
    
                string parse_id_A = ParsingDictionary.MMTermToString(planeOMA.arguments[0]);
                A_uri = ParsingDictionary.parseTermsToId[parse_id_A];
    
                OMA n = (OMA)planeOMA.arguments[1];
                normal = new Vector3(((OMF)n.arguments[0]).f, ((OMF)n.arguments[1]).f, ((OMF)n.arguments[2]).f);
                Debug.Log("Norm " + normal.ToString());
    
    
            }
            // In case of parametrized plane
            else if(planeApplicant.Equals(MMTURIs.ParametrizedPlane))
            {
                Debug.Log("planeApplicant" + planeApplicant);
    
                OMA pointAOMA = (OMA)planeOMA.arguments[0];
                string parse_id_A = ParsingDictionary.MMTermToString(planeOMA.arguments[0]);
                A_uri = ParsingDictionary.parseTermsToId[parse_id_A];
    
    
                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;
    
                Debug.Log("Para "+normal.ToString());
    
            }
            else {
                Debug.Log("planeApplicant" + planeApplicant);
                Debug.Log("?? " + MMTURIs.pointNormalPlane);
    
                return null;
            }
    
    
    
            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()
        {
    
            return  "○"+_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)
        {
            PointFact p1 = _Facts[p1URI] as PointFact;
            PointFact p2 = _Facts[p2URI] as PointFact;
    
    
            List<MMTTerm> normalArgs = new List<MMTTerm>
            {
                new OMF(normal.x),
                new OMF(normal.y),
                new OMF(normal.z)
            };
            OMA NormalVector = new OMA(new OMS(MMTURIs.Tuple), normalArgs);
    
    
    
            List<MMTTerm> planeArgs = new List<MMTTerm>
            {
                new OMS(p2URI),
                NormalVector //n
            };
    
            OMA CirclePlane = new OMA(new OMS(MMTURIs.pointNormalPlane), planeArgs);
            OMS middlePoint = new OMS(p1URI);
            OMF Radius = new OMF(radius);
    
            List<MMTTerm> outerArguments = new List<MMTTerm>
            {
               CirclePlane,
               middlePoint,
               Radius
            };
    
            //OMS constructor generates full URI
            // 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(this.Label, tp, df);
            AddFactResponse.sendAdd(mmtDecl, out this._URI);
    
    
            return mmtDecl;
        }
    
    
        /// \copydoc Fact.hasDependentFacts
        public override Boolean hasDependentFacts()
        {
            return true;
        }
    
        public Vector3 getNormal() 
        {
            return normal;
        }
    
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return  this.Pid1.GetHashCode() ^ this.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) && f1.normal == f2.normal && f1.radius == f2.radius);
        }
    }
    
    
    /// <summary>
    /// A <see cref="PointFact"/> on a <see cref="CircleFact"/>
    /// </summary>
    public class OnCircleFact : FactWrappedCRTP<OnCircleFact>
    {
        /// <summary> the point on the circle  </summary>
        public string Pid;
        /// <summary> the circle, which the point is on  </summary>
        public string Cid;
    
        /// <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;
    
            PointFact pf = _Facts[pid] as PointFact;
            CircleFact cf = _Facts[cid] as CircleFact;
            string pURI = pf.Id;
            string cURI = cf.Id;
    
            List<MMTTerm> innerArguments = new List<MMTTerm>
            {
                new OMS(cURI),
                new OMS(pURI)
            };
    
            List<MMTTerm> outerArguments = new List<MMTTerm>
            {
                new OMA(new OMS(MMTURIs.OnCircle), innerArguments)
            };
    
            //OMS constructor generates full URI
            MMTTerm tp = new OMA(new OMS(MMTURIs.Ded), outerArguments);
            MMTTerm df = null;
    
            MMTSymbolDeclaration mmtDecl = new MMTSymbolDeclaration(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="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()
        {
            return _Facts[Pid].Label + "∈" + _Facts[Cid].Label;
        }
    
        /// \copydoc Fact.hasDependentFacts
        public override Boolean hasDependentFacts()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return this.Pid.GetHashCode() ^ this.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>
    {
        /// @{ <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;
        /// @}
        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;
    
            CircleFact cf1 = _Facts[cid1] as CircleFact;
            RayFact rf2 = _Facts[rid2] as RayFact;
            MMTDeclaration mmtDecl;
            string c1URI = cf1.Id;
            string r2URI = rf2.Id;
           
            if (Mathf.Abs(Mathf.Abs(angle) - 90.0f) < 0.1) {
                // also generate an orthogonal circle line fact
                OrthogonalCircleLineFact orthofact = new OrthogonalCircleLineFact(cid1, rid2, StageStatic.stage.factState);
                bool exists;
                StageStatic.stage.factState.Add(orthofact, out exists, true );
            }
    
            mmtDecl = generateMMTDeclaration(angle, c1URI, r2URI);
    
            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;
    
            if (Mathf.Abs(Mathf.Abs(angle) - 90.0f) < 0.1)
            {
                // also generate an orthogonal circle line fact
                OrthogonalCircleLineFact orthofact = new OrthogonalCircleLineFact(Cid1, Rid2, StageStatic.stage.factState);
                bool exists;
                StageStatic.stage.factState.Add(orthofact, out exists, true);
            }
    
            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
            {
                //If dependent facts do not exist return null
                return null;
            }
        }
    
        /// \copydoc Fact.generateLabel
        protected override string generateLabel()
        {
            return  "∠" + _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()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return 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>
    {
        ///  <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;
            string c1URI = cf1.Id;
    
    
            mmtDecl = generateMMTDeclaration(c1URI,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()
        {
            return "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()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return this.Cid1.GetHashCode();
        }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        protected override bool EquivalentWrapped(RadiusFact f1, RadiusFact f2)
        {
            if (f1.Cid1 == f2.Cid1)
                return true;
    
            CircleFact c1f1 = (CircleFact)_Facts[f1.Cid1];
    
            CircleFact c1f2 = (CircleFact)_Facts[f2.Cid1];
            // if they correspond to the same circle, then automatically the radius has to be the same
    
            return (c1f1.Equivalent(c1f2));
        }
    }
    
    
    /// <summary>
    /// Area of a <see cref="CircleFact">CircleFact</see> 
    /// </summary>
    public class AreaCircleFact : FactWrappedCRTP<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) Math.PI );
    
    
            MMTDeclaration mmtDecl;
            string c1URI = cf1.Id;
    
    
            mmtDecl = generateMMTDeclaration(c1URI, 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()
        {
            return "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()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return 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>
    {
        ///  <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;
    
            CircleFact cf1 = _Facts[cid1] as CircleFact;
            PointFact pf1 = _Facts[pid1] as PointFact;
            this.vol = vol;
    
    
            MMTDeclaration mmtDecl;
            string c1URI = cf1.Id;
            string p1URI = pf1.Id;
    
    
            mmtDecl = generateMMTDeclaration(c1URI, p1URI, 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()
        {
            return "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()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return 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) && f1.vol== f2.vol );
    
        }
    }
    
    
    /// <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>
    {
        ///  <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;
    
            CircleFact cf1 = _Facts[cid1] as CircleFact;
            RayFact lf1 = _Facts[lid1] as RayFact;
    
    
            MMTDeclaration mmtDecl;
            string c1URI = cf1.Id;
            string l1URI = lf1.Id;
    
    
            mmtDecl = generateMMTDeclaration(c1URI, l1URI);
    
            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()
        {
            return   _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)
        {
            List<MMTTerm> innerArguments = new List<MMTTerm>
            {
                new OMS(c1URI),
                new OMS(l1URI)
            };
    
            List<MMTTerm> outerArguments = new List<MMTTerm>
            {
                new OMA(new OMS(MMTURIs.OrthoCircleLine), innerArguments)
            };
            MMTTerm tp = new OMA(new OMS(MMTURIs.Ded), outerArguments);
            MMTTerm df = null;
    
            return new MMTSymbolDeclaration(this.Label, tp, df);
        }
    
        /// \copydoc Fact.hasDependentFacts
        public override Boolean hasDependentFacts()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return 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>
    {
        ///  <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;
    
            CircleFact cf1 = _Facts[cid1] as CircleFact;
            CircleFact cf2 = _Facts[cid2] as CircleFact;
            this.vol = vol;
            UnEqualCirclesFact unEqualProof = _Facts[unequalproof] as UnEqualCirclesFact;
    
            MMTDeclaration mmtDecl;
            string c1URI = cf1.Id;
            string c2URI = cf2.Id;
            string pURI = unEqualProof.Id;
    
    
            mmtDecl = generateMMTDeclaration(c1URI, c2URI, vol, pURI, 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;
            this.vol = volume;
            this.proof = proof;
            this.unequalCirclesProof = unequalproof;
    
            this._URI = backendURI;
            _ = this.Label;
        }
    
        /// \copydoc Fact.parseFact(Scroll.ScrollFact)
        public new static TruncatedConeVolumeFact parseFact(Scroll.ScrollFact fact)
        {
            string uri = fact.@ref.uri;
    
            string Circle1Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[0]).uri;
            string Circle2Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[1]).uri;
            float volume = ((OMF)((Scroll.ScrollValueFact)fact).value).f;
    
            string UnEqualCirclesProof = ((OMS)(((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[2])).uri;
            OMA proof = (OMA)(((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[3]);
    
    
            if (StageStatic.stage.factState.ContainsKey(Circle1Uri) && StageStatic.stage.factState.ContainsKey(Circle2Uri))
    
                return new TruncatedConeVolumeFact(Circle1Uri, Circle2Uri, volume, UnEqualCirclesProof , proof, uri,  StageStatic.stage.factState);
    
            else    //If dependent facts do not exist return null
                return null;
        }
    
        /// \copydoc Fact.generateLabel
        protected override string generateLabel()
        {
            return "V(" + _Facts[Cid1].Label +"," + _Facts[Cid2].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="c2URI"> Uri for <see cref="Cid2"/></param>
        /// <param name="val"> <see cref="vol"/></param>
        /// <returns>struct for <see cref="AddFactResponse"/></returns>
        private MMTDeclaration generateMMTDeclaration(string c1URI, string c2URI, float val, string unequalproof, OMA proof)
        {
            MMTTerm lhs =
                new OMA(
                    new OMS(MMTURIs.TruncatedVolumeCone),
    
                    new List<MMTTerm> {
                        new OMS(c1URI),
                        new OMS(c2URI),
                        new OMS(unequalproof),
                        proof,
                    }
                );
    
            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()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return new string[] { Cid1, Cid2 };
        }
    
        /// \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.Cid2].Label;
            obj.GetComponent<FactWrapper>().fact = this;
    
            return obj;
        }
    
        /// \copydoc Fact.GetHashCode
        /// uhhh is this a problem?
        public override int GetHashCode()
        {
            return this.Cid1.GetHashCode() ^ this.Cid2.GetHashCode();
        }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        protected override bool EquivalentWrapped(TruncatedConeVolumeFact f1, TruncatedConeVolumeFact f2)
        {
            if (f1.Cid1 == f2.Cid1 && f1.Cid2 == f2.Cid2)
                return true;
    
            CircleFact c1f1 = (CircleFact)_Facts[f1.Cid1];
            CircleFact c1f2 = (CircleFact)_Facts[f2.Cid1];
    
            CircleFact c2f1 = (CircleFact)_Facts[f1.Cid2];
            CircleFact c2f2 = (CircleFact)_Facts[f2.Cid2];
    
            return (c1f1.Equivalent(c1f2) && c2f1.Equivalent(c2f2) && f1.vol == f2.vol);
    
        }
    }
    
    
    /// <summary>
    /// A RightAngleFact defined by 3  <see cref="PointFact">Pointfact</see> 
    /// </summary>
    public class RightAngleFact : FactWrappedCRTP<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;
    
            PointFact pf1 = _Facts[pid1] as PointFact;
            PointFact pf2 = _Facts[pid2] as PointFact;
            PointFact pf3 = _Facts[pid3] as PointFact;
    
    
            MMTDeclaration mmtDecl;
            string p1URI = pf1.Id;
            string p2URI = pf2.Id;
            string p3URI = pf3.Id;
    
    
            mmtDecl = generateMMTDeclaration(p1URI, p2URI, p3URI);
    
            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 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()
        {
            return _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)
        {
            List<MMTTerm> innerArguments = new List<MMTTerm>
            {
                new OMS(p1URI),
                new OMS(p2URI),
                new OMS(p3URI)
            };
    
            List<MMTTerm> outerArguments = new List<MMTTerm>
            {
                new OMA(new OMS(MMTURIs.RightAngle), innerArguments)
            };
            MMTTerm tp = new OMA(new OMS(MMTURIs.Ded), outerArguments);
            MMTTerm df = null;
    
            return new MMTSymbolDeclaration(this.Label, tp, df);
        }
    
        /// \copydoc Fact.hasDependentFacts
        public override Boolean hasDependentFacts()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return 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()
        {
            return 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>
    /// The volume of a cylinder 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 CylinderVolumeFact : FactWrappedCRTP<CylinderVolumeFact>
    {
        ///  <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 the cylinder as a float </summary>
        public float vol;
        /// <summary> a proof that both circles have the same size </summary>
        public string equalCirclesProof;
        ///  <summary> OMA proof that the two circles are parallel  </summary>
        public OMA proof;
    
        /// <summary> \copydoc Fact.Fact </summary>
        public CylinderVolumeFact() : base()
        {
            this.Cid1 = null;
            this.Cid2 = null;
            this.vol = 0.0f;
            this.proof = null;
            this.equalCirclesProof = 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 CylinderVolumeFact(CylinderVolumeFact 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.equalCirclesProof], 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 CylinderVolumeFact(string cid1, string cid2, float vol, string eqProof, OMA proof, FactOrganizer organizer) : base(organizer)
        {
            init(cid1, cid2, vol, eqProof, 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 eqProof, OMA proof)
        {
            this.Cid1 = cid1;
            this.Cid2 = cid2;
            this.proof = proof;
            this.equalCirclesProof = eqProof;
    
            CircleFact cf1 = _Facts[cid1] as CircleFact;
            CircleFact cf2 = _Facts[cid2] as CircleFact;
            EqualCirclesFact pf1 = _Facts[eqProof] as EqualCirclesFact;
            this.vol = vol;
    
    
            MMTDeclaration mmtDecl;
            string c1URI = cf1.Id;
            string c2URI = cf2.Id;
            string p1Uri = pf1.Id;
    
    
            mmtDecl = generateMMTDeclaration(c1URI, c2URI, vol,p1Uri, 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 CylinderVolumeFact(string Cid1, string Cid2, float volume, string eqProof, OMA proof, string backendURI, FactOrganizer organizer) : base(organizer)
        {
            this.Cid1 = Cid1;
            this.Cid2 = Cid2;
            this.vol = volume;
            this.proof = proof;
            this.equalCirclesProof = eqProof;
    
            this._URI = backendURI;
            _ = this.Label;
        }
    
        /// \copydoc Fact.parseFact(Scroll.ScrollFact)
        public new static CylinderVolumeFact parseFact(Scroll.ScrollFact fact)
        {
            string uri = fact.@ref.uri;
    
            string Circle1Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[0]).uri;
            string Circle2Uri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[1]).uri;
            float volume = ((OMF)((Scroll.ScrollValueFact)fact).value).f;
            string EqualCirclesProof = ((OMS)(((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[2])).uri;
    
    
    
            OMA proof = (OMA)(((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[3]);
    
            if (StageStatic.stage.factState.ContainsKey(Circle1Uri) && StageStatic.stage.factState.ContainsKey(Circle2Uri))
    
                return new CylinderVolumeFact(Circle1Uri, Circle2Uri, volume, EqualCirclesProof, proof, uri, StageStatic.stage.factState);
    
            else    //If dependent facts do not exist return null
                return null;
        }
    
        /// \copydoc Fact.generateLabel
        protected override string generateLabel()
        {
            return "V(" + _Facts[Cid1].Label + "," + _Facts[Cid2].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="c2URI"> Uri for <see cref="Cid2"/></param>
        /// <param name="val"> <see cref="vol"/></param>
        /// <returns>struct for <see cref="AddFactResponse"/></returns>
        private MMTDeclaration generateMMTDeclaration(string c1URI, string c2URI, float val, string p1URI, OMA proof)
        {
            MMTTerm lhs =
                new OMA(
                    new OMS(MMTURIs.CylinderVolume),
    
                    new List<MMTTerm> {
                        new OMS(c1URI),
                        new OMS(c2URI),
                        new OMS(p1URI),
                        proof,
                    }
                );
    
            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()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return new string[] { Cid1, Cid2, equalCirclesProof };
        }
    
        /// \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.Cid2].Label;
            obj.GetComponent<FactWrapper>().fact = this;
    
            return obj;
        }
    
        /// \copydoc Fact.GetHashCode
        /// uhhh is this a problem?
        public override int GetHashCode()
        {
            return this.Cid1.GetHashCode() ^ this.Cid2.GetHashCode();
        }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        protected override bool EquivalentWrapped(CylinderVolumeFact f1, CylinderVolumeFact f2)
        {
            if (f1.Cid1 == f2.Cid1 && f1.Cid2 == f2.Cid2)
                return true;
    
            CircleFact c1f1 = (CircleFact)_Facts[f1.Cid1];
            CircleFact c1f2 = (CircleFact)_Facts[f2.Cid1];
    
            CircleFact c2f1 = (CircleFact)_Facts[f1.Cid2];
            CircleFact c2f2 = (CircleFact)_Facts[f2.Cid2];
    
            return (c1f1.Equivalent(c1f2) && c2f1.Equivalent(c2f2) && f1.vol == f2.vol);
    
        }
    }
    
    
    
    
    /// <summary>
    /// A fact that describes, that two circles have the same size and is comprised of two <see cref="CircleFact">CircleFacts</see> 
    /// </summary>
    public class EqualCirclesFact : FactWrappedCRTP<EqualCirclesFact>
    {
        /// @{ <summary>
        /// two circles that are meant to be equal in area
        /// </summary>
        public string Cid1, Cid2;
        /// @}
    
    
    
    
        /// <summary> \copydoc Fact.Fact </summary>
        public EqualCirclesFact() : base()
        {
            this.Cid1 = null;
            this.Cid2 = 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 EqualCirclesFact(EqualCirclesFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        {
            init(old_to_new[fact.Cid1], old_to_new[fact.Cid2]);
        }
    
        /// <summary>
        /// Standard Constructor
        /// </summary>
        /// <param name="cid1">sets <see cref="Cid1"/></param>
        /// <param name="cid2">sets <see cref="Cid2"/></param>
        /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
        public EqualCirclesFact(string cid1, string cid2, FactOrganizer organizer) : base(organizer)
        {
            init(cid1, cid2);
        }
    
        /// <summary>
        /// Initiates <see cref="Cid1"/>, <see cref="Cid2"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
        /// </summary>
        /// <param name="cid1">sets <see cref="Cid1"/></param>
        /// <param name="cid2">sets <see cref="Cid2"/></param>
        private void init(string cid1, string cid2)
        {
            //TODO must be parallel lines 
            this.Cid1 = cid1;
            this.Cid2 = cid2;
    
            CircleFact cf1 = _Facts[cid1] as CircleFact;
            CircleFact cf2 = _Facts[cid2] as CircleFact;
    
            MMTDeclaration mmtDecl;
            string c1URI = cf1.Id;
            string c2URI = cf2.Id;
            mmtDecl = generateEqualCirclesFactDeclaration(c1URI, c2URI);
    
            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="backendURI">MMT URI</param>
        /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
        public EqualCirclesFact(string Cid1, string Cid2, string backendURI, FactOrganizer organizer) : base(organizer)
        {
            this.Cid1 = Cid1;
            this.Cid2 = Cid2;
    
            this._URI = backendURI;
            _ = this.Label;
        }
    
        /// \copydoc Fact.parseFact(Scroll.ScrollFact)
        public new static EqualCirclesFact parseFact(Scroll.ScrollFact fact)
        {
            OMA tp = (OMA)((Scroll.ScrollSymbolFact)fact).tp;
            if (tp == null)
                return null;
    
            string circleAUri = "";
            string circleBUri = "";
    
            string uri = fact.@ref.uri;
            OMA proof_OMA = (OMA)((Scroll.ScrollSymbolFact)fact).tp; // proof DED
    
    
            OMA parallel_circles_OMA = (OMA)proof_OMA.arguments[0]; // parallel
    
            if (parallel_circles_OMA.arguments[0] is OMS)
            {
                // Normaler Fall 
                circleAUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
                circleBUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;
    
            }
            
    
    
            if (StageStatic.stage.factState.ContainsKey(circleAUri)
             && StageStatic.stage.factState.ContainsKey(circleBUri))
    
                return new EqualCirclesFact(circleAUri, circleBUri, uri, StageStatic.stage.factState);
    
            else    //If dependent facts do not exist return null
                return null;
        }
    
        /// \copydoc Fact.generateLabel
        protected override string generateLabel()
        {
            return   _Facts[Cid1].Label + " ≠ " + _Facts[Cid2].Label;
        }
    
    
    
        /// <summary>
        /// Constructs struct for equalCirclesFact <see cref="AddFactResponse"/>
        /// </summary>
        /// <param name="c1URI"><see cref="Cid1"/></param>
        /// <param name="c2URI"><see cref="Cid2"/></param>
        /// <returns>struct for <see cref="AddFactResponse"/></returns>
        private MMTDeclaration generateEqualCirclesFactDeclaration(string c1URI, string c2URI)
        {
    
            List<MMTTerm> innerArguments = new List<MMTTerm>
            {
                new OMS(c1URI),
                new OMS(c2URI)
            };
    
            List<MMTTerm> outerArguments = new List<MMTTerm>
            {
                ///TODO change this to equal circle fact uri
                new OMA(new OMS(MMTURIs.EqualityCircles), innerArguments)
            };
    
            //OMS constructor generates full URI
            MMTTerm tp = new OMA(new OMS(MMTURIs.Ded), outerArguments);
            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()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return new string[] { Cid1, Cid2 };
        }
    
        /// \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.Cid2].Label;
            obj.GetComponent<FactWrapper>().fact = this;
            return obj;
        }
    
        /// \copydoc Fact.GetHashCode
        public override int GetHashCode()
        {
            return this.Cid1.GetHashCode() ^ this.Cid2.GetHashCode();
        }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        protected override bool EquivalentWrapped(EqualCirclesFact f1, EqualCirclesFact f2)
        {
            if ((f1.Cid1 == f2.Cid1 && f1.Cid2 == f2.Cid2))
                return true;
    
            CircleFact e1f1 = (CircleFact)_Facts[f1.Cid1];
            CircleFact e2f1 = (CircleFact)_Facts[f1.Cid2];
            CircleFact e1f2 = (CircleFact)_Facts[f2.Cid1];
            CircleFact e2f2 = (CircleFact)_Facts[f2.Cid2];
    
            return (e1f1.Equivalent(e1f2) && e2f1.Equivalent(e2f2));
        }
    }
    
    
    /// <summary>
    /// A fact that describes, that two circles have not the same size and is comprised of two <see cref="CircleFact">CircleFacts</see> 
    /// </summary>
    public class UnEqualCirclesFact : FactWrappedCRTP<UnEqualCirclesFact>
    {
        /// @{ <summary>
        /// two circles that are meant to be unequal in area
        /// </summary>
        public string Cid1, Cid2;
        /// @}
    
    
    
    
        /// <summary> \copydoc Fact.Fact </summary>
        public UnEqualCirclesFact() : base()
        {
            this.Cid1 = null;
            this.Cid2 = 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 UnEqualCirclesFact(UnEqualCirclesFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        {
            init(old_to_new[fact.Cid1], old_to_new[fact.Cid2]);
        }
    
        /// <summary>
        /// Standard Constructor
        /// </summary>
        /// <param name="cid1">sets <see cref="Cid1"/></param>
        /// <param name="cid2">sets <see cref="Cid2"/></param>
        /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
        public UnEqualCirclesFact(string cid1, string cid2, FactOrganizer organizer) : base(organizer)
        {
            init(cid1, cid2);
        }
    
        /// <summary>
        /// Initiates <see cref="Cid1"/>, <see cref="Cid2"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
        /// </summary>
        /// <param name="cid1">sets <see cref="Cid1"/></param>
        /// <param name="cid2">sets <see cref="Cid2"/></param>
        private void init(string cid1, string cid2)
        {
            //TODO must be parallel lines 
            this.Cid1 = cid1;
            this.Cid2 = cid2;
    
            CircleFact cf1 = _Facts[cid1] as CircleFact;
            CircleFact cf2 = _Facts[cid2] as CircleFact;
    
            MMTDeclaration mmtDecl;
            string c1URI = cf1.Id;
            string c2URI = cf2.Id;
            mmtDecl = generateUnEqualCirclesFactDeclaration(c1URI, c2URI);
    
            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="backendURI">MMT URI</param>
        /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
        public UnEqualCirclesFact(string Cid1, string Cid2, string backendURI, FactOrganizer organizer) : base(organizer)
        {
            this.Cid1 = Cid1;
            this.Cid2 = Cid2;
    
            this._URI = backendURI;
            _ = this.Label;
        }
    
        /// \copydoc Fact.parseFact(Scroll.ScrollFact)
        public new static UnEqualCirclesFact parseFact(Scroll.ScrollFact fact)
        {
            OMA tp = (OMA)((Scroll.ScrollSymbolFact)fact).tp;
            if (tp == null)
                return null;
    
            string circleAUri = "";
            string circleBUri = "";
    
            string uri = fact.@ref.uri;
            OMA proof_OMA = (OMA)((Scroll.ScrollSymbolFact)fact).tp; // proof DED
    
    
            OMA unequal_circles_OMA = (OMA)proof_OMA.arguments[0]; // unequal
    
            if (unequal_circles_OMA.arguments[0] is OMS)
            {
                // Normaler Fall 
                circleAUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
                circleBUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;
    
            }
    
    
    
            if (StageStatic.stage.factState.ContainsKey(circleAUri)
             && StageStatic.stage.factState.ContainsKey(circleBUri))
    
                return new UnEqualCirclesFact(circleAUri, circleBUri, uri, StageStatic.stage.factState);
    
            else    //If dependent facts do not exist return null
                return null;
        }
    
        /// \copydoc Fact.generateLabel
        protected override string generateLabel()
        {
            return _Facts[Cid1].Label + " = " + _Facts[Cid2].Label;
        }
    
    
    
        /// <summary>
        /// Constructs struct for equalCirclesFact <see cref="AddFactResponse"/>
        /// </summary>
        /// <param name="c1URI"><see cref="Cid1"/></param>
        /// <param name="c2URI"><see cref="Cid2"/></param>
        /// <returns>struct for <see cref="AddFactResponse"/></returns>
        private MMTDeclaration generateUnEqualCirclesFactDeclaration(string c1URI, string c2URI)
        {
    
            List<MMTTerm> innerArguments = new List<MMTTerm>
            {
                new OMS(c1URI),
                new OMS(c2URI)
            };
    
            List<MMTTerm> outerArguments = new List<MMTTerm>
            {
                ///TODO change this to equal circle fact uri
                new OMA(new OMS(MMTURIs.UnEqualityCircles), innerArguments)
            };
    
            //OMS constructor generates full URI
            MMTTerm tp = new OMA(new OMS(MMTURIs.Ded), outerArguments);
            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()
        {
            return true;
        }
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return new string[] { Cid1, Cid2 };
        }
    
        /// \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.Cid2].Label;
            obj.GetComponent<FactWrapper>().fact = this;
            return obj;
        }
    
        /// \copydoc Fact.GetHashCode
        public override int GetHashCode()
        {
            return this.Cid1.GetHashCode() ^ this.Cid2.GetHashCode();
        }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        protected override bool EquivalentWrapped(UnEqualCirclesFact f1, UnEqualCirclesFact f2)
        {
            if ((f1.Cid1 == f2.Cid1 && f1.Cid2 == f2.Cid2))
                return true;
    
            CircleFact e1f1 = (CircleFact)_Facts[f1.Cid1];
            CircleFact e2f1 = (CircleFact)_Facts[f1.Cid2];
            CircleFact e1f2 = (CircleFact)_Facts[f2.Cid1];
            CircleFact e2f2 = (CircleFact)_Facts[f2.Cid2];
    
            return (e1f1.Equivalent(e1f2) && e2f1.Equivalent(e2f2));
        }
    }
    
    
    
    
    
    
    /// TEST FACT
    /// 
    
    //TODO big work in progress Circle Wrapper
    
    /// TODO Work in Progress
    /// <summary>
    /// A Circle that is made out of a middle point, a plane and a radius  
    /// </summary>
    public class TestFact : FactWrappedCRTP<TestFact>
    {
    
       
    
        /// <summary> \copydoc Fact.Fact </summary>
        public TestFact() : base()
        {
            
        }
    
        /// <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 TestFact(TestFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        {
            init();
        }
    
        /// <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 TestFact( FactOrganizer organizer) : base(organizer)
        {
            init();
        }
    
        /// <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()
        {
            
    
           // mmtDecl = generateCircleFactDeclaration(p1URI, p2URI, 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 TestFact( string backendURI, FactOrganizer organizer) : base(organizer)
        {
          
            this._URI = backendURI;
            _ = this.Label;
        }
    
        /// <summary>
        /// parses the Circlefact response of the MMT-Server
        /// </summary>
        /// \copydoc Fact.parseFact(Scroll.ScrollFact) 
        public new static TestFact parseFact(Scroll.ScrollFact fact)
        {
            string uri = fact.@ref.uri;
            Debug.Log("TestFact Uri:" + uri);
            return new TestFact(uri, StageStatic.stage.factState);
          
        }
    
        /// \copydoc Fact.generateLabel
        protected override string generateLabel()
        {
    
            return "test" ;
        }
    
    
    
        /// <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)
        {
            PointFact p1 = _Facts[p1URI] as PointFact;
            PointFact p2 = _Facts[p2URI] as PointFact;
    
    
            List<MMTTerm> normalArgs = new List<MMTTerm>
            {
                new OMF(normal.x),
                new OMF(normal.y),
                new OMF(normal.z)
            };
            OMA NormalVector = new OMA(new OMS(MMTURIs.Tuple), normalArgs);
    
    
    
            List<MMTTerm> planeArgs = new List<MMTTerm>
            {
                new OMS(p2URI),
                NormalVector //n
            };
    
            OMA CirclePlane = new OMA(new OMS(MMTURIs.pointNormalPlane), planeArgs);
            OMS middlePoint = new OMS(p1URI);
            OMF Radius = new OMF(radius);
    
            List<MMTTerm> outerArguments = new List<MMTTerm>
            {
               CirclePlane,
               middlePoint,
               Radius
            };
    
            //OMS constructor generates full URI
            // 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(this.Label, tp, df);
            AddFactResponse.sendAdd(mmtDecl, out this._URI);
    
    
            return mmtDecl;
        }
    
    
        /// \copydoc Fact.hasDependentFacts
        public override Boolean hasDependentFacts()
        {
            return false;
        }
    
     
    
        /// \copydoc Fact.getDependentFactIds
        public override string[] getDependentFactIds()
        {
            return new string[] { };
        }
    
        /// \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(1).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Lid2].Label;
    
            obj.GetComponent<FactWrapper>().fact = this;
            return obj;
        }
    
        /// \copydoc Fact.GetHashCode
        public override int GetHashCode()
        {
            return 112315414;// this.Pid1.GetHashCode() ^ this.Pid2.GetHashCode();
        }
    
        /// \copydoc Fact.Equivalent(Fact, Fact)
        protected override bool EquivalentWrapped(TestFact f1, TestFact f2)
        {
            return false;
        }
    }