levels.json

{ "functions":
  { "+": "ℝ"
  , "0": "ℝ"
  , "-": "ℝ"
  , "−": "ℝ"
  , "×": "ℝ"
  , "1": "ℝ"
  , "⅟": "ℝ"
  , "÷": "ℝ"
  , "^": "ℝ"
  , "√": "ℝ"
  , "ln": "ℝ"
  , "-1": "ℝ"
  , "2": "ℝ"
  , "3": "ℝ"
  , "4": "ℝ"
  , "½": "ℝ"
  , "⅓": "ℝ"
  , "ℯ": "ℝ"
  , "¬": "Ω"
  , "<": "Ω"
  , "≤": "Ω"
  , "≠": "Ω"
  }
, "levels":
  { "Intro.0":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["=",[0,0]], [ 0,3]]
      ]
    , "hypotheses": []
    , "conclusion": 1
    , "_comment": "Equality"
    , "map_position": [0,0]
    , "bezier_vector": [1,0]
    , "prereqs": []
    , "next_level": ["Intro.2", "Intro.3"]
    }
  , "Intro.2":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["=",[0,1]], [-1,3]]
      , [["=",[1,0]], [ 1,3]]
      ]
    , "hypotheses": [2]
    , "conclusion": 3
    , "_comment" : "Symmetry of equality."
    , "map_position": [1.5,-0.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["Intro.0"]
    , "next_level": ["Intro.3", "Intro.1"]
    }
  , "Intro.3":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a",         [-1.5,0]]
      , ["b",         [ 0  ,0]]
      , ["c",         [ 1.5,0]]
      , [["=",[0,1]], [-1.5,3]]
      , [["=",[1,2]], [ 1.5,3]]
      , [["=",[0,2]], [ 0  ,3]]
      ]
    , "hypotheses": [3,4]
    , "conclusion": 5
    , "_comment" : "Transitivity of equality."
    , "map_position": [1.5,0.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["Intro.0"]
    , "next_level": ["Intro.2", "Intro.1"]
    }
  , "Intro.1":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["a",         [ 1,0]]
      , [["=",[0,1]], [ 0,3]]
      ]
    , "hypotheses": []
    , "conclusion": 2
    , "map_position": [3,0]
    , "bezier_vector": [1,0]
    , "text_box": ["Drag equal nodes together."]
    , "prereqs": ["Intro.2", "Intro.3"]
    , "next_level": ["Intro.4"]
    }
  , "Intro.4":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a",         [-1.5,0]]
      , ["b",         [ 0  ,0]]
      , ["c",         [ 1.5,0]]
      , [["+",[0,1]], [-1.5,3]]
      , [["=",[1,2]], [ 1.5,3]]
      , [["+",[0,2]], [ 0  ,3]]
      , [["=",[3,5]], [-0.75,5.5]]
      ]
    , "hypotheses": [4]
    , "conclusion": 6
    , "map_position": [4.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["Intro.1"]
    , "next_level": ["+-id"]
    }


  , "+-id":
    { "axiom": true
    , "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [-1  ,0]]
      , [["0",[   ]], [ 1  ,0]]
      , [["+",[0,1]], [ 0.5,2.5]]
      , [["=",[0,2]], [-0.5,4.5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [9,0]
    , "bezier_vector": [1,0]
    , "text_box": ["Axiom: You don't need to solve this."]
    , "prereqs": ["Intro.4"]
    , "next_level": ["+.0"]
    }
  , "+.0":
    { "unlocks": ["theorem-application", "cases"]
    , "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [-1.5,0]]
      , ["a"        , [ 0  ,0]]
      , [["0",[   ]], [ 1.5,0]]
      , [["+",[1,2]], [ 0.75,2.5]]
      , [["=",[0,3]], [ 0  ,5]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [10.5,0]
    , "bezier_vector": [2.5,0]
    , "text_box": ["New feature! Import the axiom to solve this one.", "ThmApp"]
    , "prereqs": ["+-id"]
    , "next_level": ["+.1"]
    }
  , "+.1":
    { "variables": {}
    , "nodes":
      [ [["0",[   ]], [-1.5,0]]
      , [["0",[   ]], [ 0  ,0]]
      , [["0",[   ]], [ 1.5,0]]
      , [["+",[0,1]], [-0.75,2.5]]
      , [["=",[3,2]], [ 0  ,5]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [12,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["+.0"]
    , "next_level": ["+-comm", "+-assoc", "+-inv"]
    }
  , "+-comm":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-1  ,0]]
      , ["b"        , [ 1  ,0]]
      , [["+",[0,1]], [-1  ,3]]
      , [["+",[1,0]], [ 1  ,3]]
      , [["=",[2,3]], [ 0  ,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [13.5,-1.5]
    , "bezier_vector": [1,0]
    , "text_box": ["Another axiom."]
    , "prereqs": ["+.1"]
    , "next_level": ["+.2", "+.3"]
    }
  , "+-assoc":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-1.5,0]]
      , ["b"        , [ 0  ,0]]
      , ["c"        , [ 1.5,0]]
      , [["+",[0,1]], [-1.5,3.25]]
      , [["+",[1,2]], [ 1.5,3.25]]
      , [["+",[3,2]], [-0.75,5]]
      , [["+",[0,4]], [ 0.75,5]]
      , [["=",[5,6]], [ 0  ,7]]
      ]
    , "hypotheses": []
    , "conclusion": 7
    , "map_position": [13.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["+.1"]
    , "next_level": ["+.5", "+.3"]
    }
  , "+-inv":
    { "axiom": true
    , "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [-1  ,0]]
      , [["-",[0  ]], [ 1  ,2]]
      , [["+",[0,1]], [ 0  ,4]]
      , [["0",[   ]], [ 2  ,4]]
      , [["=",[2,3]], [ 1  ,6]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [13.5,1.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["+.1"]
    , "next_level": ["+.5"]
    }
  , "+.2":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [ 1  ,0]]
      , [["0",[   ]], [-1  ,0]]
      , [["+",[1,0]], [-0.5,2.5]]
      , [["=",[2,0]], [ 0.5,4.5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [15,-2.25]
    , "bezier_vector": [1,0]
    , "prereqs": ["+-comm"]
    , "next_level": ["+.7"]
    }
  , "+.3":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-5,0]]
      , ["b"        , [-3,0]]
      , ["c"        , [-1,0]]
      , [["+",[0,1]], [-4,3]]
      , [["+",[3,2]], [-3,6]]
      , ["a"        , [ 1,0]]
      , ["c"        , [ 3,0]]
      , ["b"        , [ 5,0]]
      , [["+",[5,6]], [ 2,3]]
      , [["+",[8,7]], [ 3,6]]
      , [["=",[4,9]], [ 0,9]]
      ]
    , "hypotheses": []
    , "conclusion": 10
    , "map_position": [15,-0.75]
    , "bezier_vector": [1,0]
    , "prereqs": ["+-assoc", "+-comm"]
    , "next_level": ["+.4"]
    }
  , "+.4":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ", "d": "ℝ"}
    , "nodes":
      [ ["a"        , [-3,0]]
      , ["b"        , [-1,0]]
      , ["c"        , [ 1,0]]
      , ["d"        , [ 3,0]]
      , [["+",[0,1]], [-3,3]]
      , [["+",[0,2]], [-1,3]]
      , [["+",[1,3]], [ 1,3]]
      , [["+",[2,3]], [ 3,3]]
      , [["+",[4,7]], [-1,6]]
      , [["+",[5,6]], [ 1,6]]
      , [["=",[8,9]], [ 0,9]]
      ]
    , "hypotheses": []
    , "conclusion": 10
    , "map_position": [18,-2.25]
    , "bezier_vector": [1,0]
    , "prereqs": ["+.3"]
    , "next_level": []
    }
  , "+.5":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-2  ,0]]
      , ["b",         [ 0  ,0]]
      , [["+",[0,1]], [-1  ,3]]
      , [["-",[1  ]], [ 1  ,3]]
      , [["+",[2,3]], [ 0  ,5]]
      , [["=",[0,4]], [-1  ,7]]
      ]
    , "hypotheses": []
    , "conclusion": 5
    , "map_position": [15,0.75]
    , "bezier_vector": [1,0]
    , "prereqs": ["+-inv", "+-assoc"]
    , "next_level": ["+.6"]
    }
  , "+.6":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a",         [-1.5,0]]
      , ["b",         [ 0  ,0]]
      , ["c",         [ 1.5,0]]
      , [["=",[0,1]], [-1.5,3]]
      , [["+",[0,2]], [ 0  ,3]]
      , [["+",[1,2]], [ 1.5,3]]
      , [["=",[4,5]], [ 0.75,5.5]]
      ]
    , "hypotheses": [6]
    , "conclusion": 3
    , "map_position": [16.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["+.5", "+.3"]
    , "next_level": ["+.7", "+.8"]
    }
  , "+.7":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["a",         [ 1,0]]
      , [["-",[1  ]], [ 1,2]]
      , [["-",[2  ]], [ 1,4]]
      , [["=",[0,3]], [ 0,6]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [18,0.75]
    , "bezier_vector": [1,0]
    , "prereqs": ["+.6"]
    , "next_level": []
    }
  , "+.8":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ [["0",[   ]], [-1,0]]
      , [["0",[   ]], [ 1,0]]
      , [["-",[1  ]], [ 1,2]]
      , [["=",[0,2]], [ 0,4]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [18,-0.75]
    , "bezier_vector": [1,0]
    , "prereqs": ["+.6", "+.2"]
    , "next_level": []
    }


  , "−-def":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["-",[1  ]], [ 1,2.5]]
      , [["+",[0,2]], [ 1,5]]
      , [["−",[0,1]], [-1,3.5]]
      , [["=",[4,3]], [ 0,7]]
      ]
    , "hypotheses": []
    , "conclusion": 5
    , "map_position": [22.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["+.7", "+.8", "+.4"]
    , "next_level": ["−.0"]
    }
  , "−.0":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["-",[1  ]], [ 1,2.5]]
      , [["−",[0,2]], [ 1,5]]
      , [["+",[0,1]], [-1,3.5]]
      , [["=",[4,3]], [ 0,7]]
      ]
    , "hypotheses": []
    , "conclusion": 5
    , "map_position": [24,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["−-def"]
    , "next_level": ["−.1"]
    }
  , "−.1":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , [["−",[0,0]], [-1,2.5]]
      , [["0",[   ]], [ 1,2.5]]
      , [["=",[1,2]], [ 0,5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [25.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["−.0"]
    , "next_level": []
    }


   , "×-id":
    { "axiom": true
    , "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [-1  ,0]]
      , [["1",[   ]], [ 1  ,0]]
      , [["×",[0,1]], [ 0.5,2.5]]
      , [["=",[0,2]], [-0.5,4.5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [31.5,-1]
    , "bezier_vector": [1,0]
    , "prereqs": ["−.1"]
    , "next_level": ["×-comm", "×-assoc", "×.0"]
    }
   , "×-comm":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-1  ,0]]
      , ["b"        , [ 1  ,0]]
      , [["×",[0,1]], [-1  ,3]]
      , [["×",[1,0]], [ 1  ,3]]
      , [["=",[2,3]], [ 0  ,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [31.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["−.1"]
    , "next_level": ["×-assoc", "×-id", "×.0", "×.1"]
    }
  , "×-assoc":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-1.5,0]]
      , ["b"        , [ 0  ,0]]
      , ["c"        , [ 1.5,0]]
      , [["×",[0,1]], [-1.5,3.25]]
      , [["×",[1,2]], [ 1.5,3.25]]
      , [["×",[3,2]], [-0.75,5]]
      , [["×",[0,4]], [ 0.75,5]]
      , [["=",[5,6]], [ 0  ,7]]
      ]
    , "hypotheses": []
    , "conclusion": 7
    , "map_position": [31.5,1]
    , "bezier_vector": [1,0]
    , "prereqs": ["−.1"]
    , "next_level": ["×-id", "×-comm", "×.1"]
    }
  , "×.0":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [ 1  ,0]]
      , [["1",[   ]], [-1  ,0]]
      , [["×",[1,0]], [-0.5,2.5]]
      , [["=",[2,0]], [ 0.5,4.5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [33,-0.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["×-id", "×-comm"]
    , "next_level": ["×.1", "×-distrib-+"]
    }
  , "×.1":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-5,0]]
      , ["b"        , [-3,0]]
      , ["c"        , [-1,0]]
      , [["×",[0,1]], [-4,3]]
      , [["×",[3,2]], [-3,6]]
      , ["a"        , [ 1,0]]
      , ["c"        , [ 3,0]]
      , ["b"        , [ 5,0]]
      , [["×",[5,6]], [ 2,3]]
      , [["×",[8,7]], [ 3,6]]
      , [["=",[4,9]], [ 0,9]]
      ]
    , "hypotheses": []
    , "conclusion": 10
    , "map_position": [33,0.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["×-assoc", "×-comm"]
    , "next_level": ["×.0", "×-distrib-+"]
    }
  , "×-distrib-+":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-1.5,0]]
      , ["b"        , [ 0  ,0]]
      , ["c"        , [ 1.5,0]]
      , [["×",[0,1]], [-1.5,3]]
      , [["×",[0,2]], [ 0  ,3]]
      , [["+",[1,2]], [ 1.5,3]]
      , [["+",[3,4]], [-0.75,5.5]]
      , [["×",[0,5]], [ 0.75,5.5]]
      , [["=",[6,7]], [ 0  ,8]]
      ]
    , "hypotheses": []
    , "conclusion": 8
    , "map_position": [34.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["×.0", "×.1"]
    , "next_level": ["×.2", "×.4"]
    }
  , "×.2":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-1.5,0]]
      , ["b"        , [ 0  ,0]]
      , ["c"        , [ 1.5,0]]
      , [["+",[0,1]], [-1.5,3]]
      , [["×",[0,2]], [ 0  ,3]]
      , [["×",[1,2]], [ 1.5,3]]
      , [["×",[3,2]], [-0.75,5.5]]
      , [["+",[4,5]], [ 0.75,5.5]]
      , [["=",[6,7]], [ 0  ,8]]
      ]
    , "hypotheses": []
    , "conclusion": 8
    , "map_position": [36,-0.75]
    , "bezier_vector": [1,0]
    , "prereqs": ["×-distrib-+"]
    , "next_level": ["×.3"]
    }
  , "×.3":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ", "d": "ℝ"}
    , "nodes":
      [ ["a"        , [-7,0]]
      , ["b"        , [-5,0]]
      , ["c"        , [-3,0]]
      , ["d"        , [-1,0]]
      , [["+",[0,1]], [-6,3]]
      , [["+",[2,3]], [-2,3]]
      , [["×",[4,5]], [-4,6]]

      , ["a"          , [ 2  ,0]]
      , ["c"          , [ 3  ,0]]
      , [["×",[ 7, 8]], [ 2.5,2]]
      , ["a"          , [ 4.5,0]]
      , ["d"          , [ 5.5,0]]
      , [["×",[10,11]], [ 5  ,2]]
      , ["b"          , [ 7  ,0]]
      , ["c"          , [ 8  ,0]]
      , [["×",[13,14]], [ 7.5,2]]
      , ["b"          , [ 9.5,0]]
      , ["d"          , [10.5,0]]
      , [["×",[16,17]], [10  ,2]]
      , [["+",[ 9,12]], [3.75,4]]
      , [["+",[15,18]], [8.75,4]]
      , [["+",[19,20]], [6.25,6]]
      , [["=",[6,21]], [1.125,9]]
      ]
    , "hypotheses": []
    , "conclusion": 22
    , "map_position": [37.5,-0.75]
    , "bezier_vector": [1,0]
    , "text_box": ["F.O.I.L."]
    , "prereqs": ["×.2"]
    , "next_level": ["×.6"]
    }
  , "×.4":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [-1  ,0]]
      , [["0",[   ]], [ 1  ,0]]
      , [["×",[0,1]], [-0.5,2.5]]
      , [["=",[2,1]], [ 0.5,4.5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [36,0.75]
    , "bezier_vector": [1,0]
    , "prereqs": ["×-distrib-+"]
    , "next_level": ["×.5"]
    }
  , "×.5":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-1  ,0]]
      , ["b"        , [ 1  ,0]]
      , [["-",[1  ]], [ 1  ,2]]
      , [["×",[0,1]], [-1  ,3]]
      , [["-",[3  ]], [-1  ,5.5]]
      , [["×",[0,2]], [ 1  ,4.5]]
      , [["=",[4,5]], [ 0  ,7.5]]
      ]
    , "hypotheses": []
    , "conclusion": 6
    , "map_position": [37.5,0.75]
    , "bezier_vector": [1,0]
    , "prereqs": ["×.4"]
    , "next_level": ["×.6"]
    }
  , "×.6":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-3.5,0]]
      , ["b"        , [-1.5,0]]
      , [["+",[0,1]], [-3.5,3]]
      , [["−",[0,1]], [-1.5,3]]
      , [["×",[2,3]], [-2.5,6]]

      , ["a"        , [ 1.5,0]]
      , ["b"        , [ 3.5,0]]
      , [["×",[5,5]], [ 1.5,3]]
      , [["×",[6,6]], [ 3.5,3]]
      , [["−",[7,8]], [ 2.5,6]]
      , [["=",[4,9]], [ 0,9]]
      ]
    , "hypotheses": []
    , "conclusion": 10
    , "map_position": [39,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["×.3", "×.5"]
    , "next_level": []
    }


  , "-1-def":
    { "axiom" : true
    , "variables": {}
    , "nodes":
      [ [["1",[   ]], [-1,0]]
      , [["-",[0  ]], [-1,3]]
      , [["-1",[  ]], [ 1,3]]
      , [["=",[1,2]], [ 0,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [43.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["×.6"]
    , "next_level": ["-1×"]
    }
  , "-1×":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ [["-1",[  ]], [-2,0]]
      , ["a"        , [ 0,0]]
      , [["×",[0,1]], [-1,3]]
      , [["-",[1  ]], [ 1,3]]
      , [["=",[2,3]], [ 0,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [45,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["-1-def"]
    , "next_level": ["-1×-1"]
    }
  , "-1×-1":
    { "variables": {}
    , "nodes":
      [ [["-1",[  ]], [-1,0]]
      , [["×",[0,0]], [-1,3]]
      , [["1",[   ]], [ 1,3]]
      , [["=",[1,2]], [ 0,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [46.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["-1×"]
    , "next_level": ["-1<0"]
    }
  , "-1<0":
    { "variables": {}
    , "nodes":
      [ [["-1",[  ]], [-1,0]]
      , [["0",[   ]], [ 1,0]]
      , [["<",[0,1]], [ 0,3]]
      ]
    , "hypotheses": []
    , "conclusion": 2
    , "map_position": [48,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["-1×-1", "0<1"]
    , "next_level": ["2-def"]
    }
  , "2-def":
    { "axiom" : true
    , "variables": {}
    , "nodes":
      [ [["1",[   ]], [-2,0]]
      , [["1",[   ]], [ 0,0]]
      , [["+",[0,1]], [-1,3]]
      , [["2",[   ]], [ 1,3]]
      , [["=",[2,3]], [ 0,6]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [51,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["×.6"]
    , "next_level": ["3-def"]
    }
  , "3-def":
    { "axiom" : true
    , "variables": {}
    , "nodes":
      [ [["2",[   ]], [-2,0]]
      , [["1",[   ]], [ 0,0]]
      , [["+",[0,1]], [-1,3]]
      , [["3",[   ]], [ 1,3]]
      , [["=",[2,3]], [ 0,6]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [52,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["2-def"]
    , "next_level": ["4-def"]
    }
  , "4-def":
    { "axiom" : true
    , "variables": {}
    , "nodes":
      [ [["3",[   ]], [-2,0]]
      , [["1",[   ]], [ 0,0]]
      , [["+",[0,1]], [-1,3]]
      , [["4",[   ]], [ 1,3]]
      , [["=",[2,3]], [ 0,6]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [53,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["3-def"]
    , "next_level": ["0<2"]
    }
  , "0<2":
    { "variables": {}
    , "nodes":
      [ [["0",[  ]], [-1,0]]
      , [["2",[   ]], [ 1,0]]
      , [["<",[0,1]], [ 0,3]]
      ]
    , "hypotheses": []
    , "conclusion": 2
    , "map_position": [54.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["4-def"]
    , "next_level": ["½-def"]
    }
  , "½-def":
    { "axiom" : true
    , "variables": {}
    , "nodes":
      [ [["2",[   ]], [-1,0]]
      , [["⅟",[0  ]], [-1,3]]
      , [["½",[   ]], [ 1,3]]
      , [["=",[1,2]], [ 0,6]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [56,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["0<2"]
    , "next_level": ["⅓-def"]
    }
  , "⅓-def":
    { "axiom" : true
    , "variables": {}
    , "nodes":
      [ [["3",[   ]], [-1,0]]
      , [["⅟",[0  ]], [-1,3]]
      , [["⅓",[   ]], [ 1,3]]
      , [["=",[1,2]], [ 0,6]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [57,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["½-def"]
    , "next_level": ["0<½"]
    }
  , "0<½":
    { "variables": {}
    , "nodes":
      [ [["0",[  ]], [-1,0]]
      , [["½",[   ]], [ 1,0]]
      , [["<",[0,1]], [ 0,3]]
      ]
    , "hypotheses": []
    , "conclusion": 2
    , "map_position": [58.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["⅓-def", "⅟.2"]
    , "next_level": ["½+½"]
    }
  , "½+½":
    { "variables": {}
    , "nodes":
      [ [["½",[   ]], [-1,0]]
      , [["+",[0,0]], [-1,3]]
      , [["1",[   ]], [ 1,0]]
      , [["=",[1,2]], [ 0,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [60,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["0<½"]
    , "next_level": ["½−⅓"]
    }
  , "½−⅓":
    { "variables": {}
    , "nodes":
      [ [["½",[   ]], [-1,0]]
      , [["⅓",[   ]], [ 1,0]]
      , [["−",[0,1]], [-1,3]]
      , [["×",[0,1]], [ 1,3]]
      , [["=",[2,3]], [ 0,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [73.5,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["½+½", "÷.5"]
    , "next_level": ["1½×1⅓"]
    }
  , "1½×1⅓":
    { "variables": {}
    , "nodes":
      [ [["1",[   ]], [-3,0]]
      , [["½",[   ]], [-1,0]]
      , [["1",[   ]], [ 1,0]]
      , [["⅓",[   ]], [ 3,0]]
      , [["+",[0,1]], [-2,3]]
      , [["+",[2,3]], [ 2,3]]
      , [["×",[4,5]], [ 0,6]]
      , [["2",[   ]], [ 2,6]]
      , [["=",[6,7]], [ 1,8.5]]
      ]
    , "hypotheses": []
    , "conclusion": 8
    , "map_position": [75,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["½−⅓"]
    , "next_level": ["2+2=4"]
    }
  , "2+2=4":
    { "variables": {}
    , "nodes":
      [ [["2",[   ]], [-2,0]]
      , [["2",[   ]], [ 0,0]]
      , [["+",[0,1]], [-1,3]]
      , [["4",[   ]], [ 1,3]]
      , [["=",[2,3]], [ 0,6]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [84,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["1½×1⅓","^1"]
    , "next_level": ["2×2=4"]
    }
  , "2×2=4":
    { "variables": {}
    , "nodes":
      [ [["2",[   ]], [-2,0]]
      , [["2",[   ]], [ 0,0]]
      , [["×",[0,1]], [-1,3]]
      , [["4",[   ]], [ 1,3]]
      , [["=",[2,3]], [ 0,6]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [85,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["2+2=4"]
    , "next_level": ["2^2=4"]
    }
  , "2^2=4":
    { "variables": {}
    , "nodes":
      [ [["2",[   ]], [-2,0]]
      , [["2",[   ]], [ 0,0]]
      , [["^",[0,1]], [-1,3]]
      , [["4",[   ]], [ 1,3]]
      , [["=",[2,3]], [ 0,6]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [86,0]
    , "bezier_vector": [1,0]
    , "prereqs": ["2×2=4"]
    , "next_level": []
    }








  , "⇒.0":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["⇒",[0,1]], [ 0,3]]
      ]
    , "hypotheses": [0,2]
    , "conclusion": 1
    , "_comment" : "Introduce implications."
    , "map_position": [1.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": []
    , "next_level": ["⇒.1"]
    , "text_box": ["New feature! Implication.", "Implication"]
    }
  , "⇒.1":
    { "variables": {"a": "Ω", "b": "Ω", "c": "Ω", "d": "Ω", "e": "Ω"}
    , "nodes":
      [ ["a",         [-4,0]]
      , ["b",         [-2,0.5625]]
      , ["c",         [ 0,0.75]]
      , ["d",         [ 2,0.5625]]
      , ["e",         [ 4,0]]
      , [["⇒",[0,1]], [-3,3]]
      , [["⇒",[1,2]], [-1,3]]
      , [["⇒",[2,3]], [ 1,3]]
      , [["⇒",[3,4]], [ 3,3]]
      ]
    , "hypotheses": [0,5,6,7,8]
    , "conclusion": 4
    , "_comment" : "Implication chain."
    , "map_position": [3,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["⇒.0"]
    , "next_level": ["⇒.2"]
    }
  , "⇒.2":
    { "variables": {"a": "Ω", "b": "Ω", "c": "Ω"}
    , "nodes":
      [ ["a",         [-2,0]]
      , ["b",         [ 0,0]]
      , ["c",         [ 2,0]]
      , [["⇒",[0,1]], [-2,3]]
      , [["⇒",[1,2]], [ 2,3]]
      , [["⇒",[0,2]], [ 0,3]]
      ]
    , "hypotheses": [3,4]
    , "conclusion": 5
    , "_comment" : "Implication introduction."
    , "map_position": [4.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["⇒.1"]
    , "next_level": ["⇒.3"]
    }
  , "⇒.3":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["⇒",[0,1]], [ 1,2]]
      , [["=",[0,2]], [-1,3]]
      , [["⇒",[3,2]], [ 0,5]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [6,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["Intro.4", "⇒.2"]
    , "next_level": ["PropExt"]
    }
  , "PropExt":
    { "axiom": true
    , "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["⇒",[0,1]], [-1.5,3]]
      , [["⇒",[1,0]], [ 0,3]]
      , [["=",[0,1]], [ 1.5,3]]
      ]
    , "hypotheses": [2,3]
    , "conclusion": 4
    , "map_position": [7.5,5]
    , "bezier_vector": [2.5,0]
    , "prereqs": ["⇒.3", "+.0"]
    , "next_level": ["⇒.4"]
    }
  , "⇒.4":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b"        , [ 1,0]]
      , [["=",[0,1]], [ 0,3]]
      ]
    , "hypotheses": [0,1]
    , "conclusion": 2
    , "map_position": [9,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["PropExt"]
    , "next_level": []
    }


  , "∧.0":
    { "variables": {"a": "Ω", "b": "Ω", "c": "Ω"}
    , "nodes":
      [ ["a",         [-2,0]]
      , ["b",         [ 0,0]]
      , ["c",         [ 2,0]]
      , [["∧",[0,1]], [-1,3]]
      , [["∧",[1,2]], [ 1,3]]
      , [["∧",[3,4]], [ 0,6]]
      ]
    , "hypotheses": [0,1,2]
    , "conclusion": 5
    , "_comment": "Conjunction introduction."
    , "map_position": [10.5,4]
    , "bezier_vector": [1,0]
    , "prereqs": ["⇒.4"]
    , "next_level": ["∧.1"]
    , "text_box": ["New feature! Conjunction.", "Conjunction"]
    }
  , "∧.1":
    { "variables": {"a": "Ω", "b": "Ω", "c": "Ω"}
    , "nodes":
      [ ["a",         [-2,0]]
      , ["b",         [ 0,0]]
      , ["c",         [ 2,0]]
      , [["∧",[0,1]], [-1,3]]
      , [["∧",[1,2]], [ 1,3]]
      , [["∧",[3,4]], [ 0,6]]
      ]
    , "hypotheses": [5]
    , "conclusion": 1
    , "_comment" : "Conjunction elimination."
    , "map_position": [12,4]
    , "bezier_vector": [1,0]
    , "prereqs": ["∧.0"]
    , "next_level": ["∧.2"]
    }
  , "∧.2":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["∧",[0,1]], [-1,3]]
      , [["∧",[1,0]], [ 1,3]]
      ]
    , "hypotheses": [2]
    , "conclusion": 3
    , "map_position": [13.5,4]
    , "bezier_vector": [1,0]
    , "prereqs": ["∧.1"]
    , "next_level": ["∧.3"]
    }
  , "∧.3":
    { "variables": {"a": "Ω", "b": "Ω", "c": "Ω"}
    , "nodes":
      [ ["a"        , [-1.5,0]]
      , ["b"        , [ 0  ,0]]
      , ["c"        , [ 1.5,0]]
      , [["∧",[0,1]], [-1.5,3.25]]
      , [["∧",[1,2]], [ 1.5,3.25]]
      , [["∧",[3,2]], [-0.75,5]]
      , [["∧",[0,4]], [ 0.75,5]]
      , [["=",[5,6]], [ 0  ,7]]
      ]
    , "hypotheses": []
    , "conclusion": 7
    , "map_position": [15,4]
    , "bezier_vector": [1,0]
    , "prereqs": ["∧.2"]
    , "next_level": ["∧.4"]
    }
  , "∧.4":
    { "variables": {"a": "Ω"}
    , "nodes":
      [ [["∧",[]   ], [-1.25,0]]
      , ["a",         [ 1.25,0]]
      , [["∧",[0]  ], [-2,3]]
      , [["∧",[0,1]], [ 0,3]]
      , [["∧",[1]  ], [ 2,3]]
      , [["∧",[2,3,4]],[0,6]]
      ]
    , "hypotheses": [1]
    , "conclusion": 5
    , "_comment" : "n-ary disjunctions"
    , "map_position": [16.5,4]
    , "bezier_vector": [1,0]
    , "prereqs": ["∧.3"]
    , "next_level": ["∧.5"]
    }
  , "∧.5":
    { "variables": {}
    , "nodes":
      [ [["∧",[]], [0,0]]
      ]
    , "hypotheses": []
    , "conclusion": 0
    , "map_position": [18,4]
    , "bezier_vector": [1,0]
    , "prereqs": ["∧.4"]
    , "next_level": ["⇒.7"]
    }


  , "∨.0":
    { "variables": {"a": "Ω", "b": "Ω", "c": "Ω"}
    , "nodes":
      [ ["a",         [-2,0]]
      , ["b",         [ 0,0]]
      , ["c",         [ 2,0]]
      , [["∨",[0,1]], [-1,3]]
      , [["∨",[1,2]], [ 1,3]]
      , [["∨",[3,4]], [ 0,6]]
      ]
    , "hypotheses": [1]
    , "conclusion": 5
    , "_comment": "Disjunction introduction."
    , "map_position": [10.5,6]
    , "bezier_vector": [1,0]
    , "prereqs": ["⇒.4"]
    , "next_level": ["∨.1"]
    , "text_box": ["New feature! Disjunction.", "Disjunction"]
    }
  , "∨.1":
    { "variables": {"a": "Ω"}
    , "nodes":
      [ ["a",         [-2,0]]
      , ["a",         [ 0,0]]
      , ["a",         [ 2,0]]
      , [["∨",[0,1]], [-1,3]]
      , [["∨",[1,2]], [ 1,3]]
      , [["∨",[3,4]], [ 0,6]]
      ]
    , "hypotheses": [5]
    , "conclusion": 1
    , "_comment" : "Disjunction elimination."
    , "map_position": [12,6]
    , "bezier_vector": [1,0]
    , "prereqs": ["∨.0"]
    , "next_level": ["∨.2"]
    }
  , "∨.2":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["∨",[0,1]], [-1,3]]
      , [["∨",[1,0]], [ 1,3]]
      ]
    , "hypotheses": [2]
    , "conclusion": 3
    , "map_position": [13.5,6]
    , "bezier_vector": [1,0]
    , "prereqs": ["∨.1"]
    , "next_level": ["∨.3"]
    }
  , "∨.3":
    { "variables": {"a": "Ω", "b": "Ω", "c": "Ω"}
    , "nodes":
      [ ["a"        , [-1.5,0]]
      , ["b"        , [ 0  ,0]]
      , ["c"        , [ 1.5,0]]
      , [["∨",[0,1]], [-1.5,3.25]]
      , [["∨",[1,2]], [ 1.5,3.25]]
      , [["∨",[3,2]], [-0.75,5]]
      , [["∨",[0,4]], [ 0.75,5]]
      , [["=",[5,6]], [ 0  ,7]]
      ]
    , "hypotheses": []
    , "conclusion": 7
    , "map_position": [15,6]
    , "bezier_vector": [1,0]
    , "prereqs": ["∨.2"]
    , "next_level": ["∨.4"]
    }
  , "∨.4":
    { "variables": {"a": "Ω"}
    , "nodes":
      [ [["∨",[]   ], [-1.25,0]]
      , ["a",         [ 1.25,0]]
      , [["∨",[0]  ], [-2,3]]
      , [["∨",[0,1]], [ 0,3]]
      , [["∨",[1]  ], [ 2,3]]
      , [["∨",[2,3,4]],[0,6]]
      ]
    , "hypotheses": [5]
    , "conclusion": 1
    , "_comment" : "n-ary disjunctions"
    , "map_position": [16.5,6]
    , "bezier_vector": [1,0]
    , "prereqs": ["∨.3"]
    , "next_level": ["∨.5"]
    }
  , "∨.5":
    { "variables": {"a": "Ω"}
    , "nodes":
      [ [["∨",[]],    [-1,0]]
      , ["a",         [ 1,0]]
      ]
    , "hypotheses": [0]
    , "conclusion": 1
    , "map_position": [18,6]
    , "bezier_vector": [1,0]
    , "prereqs": ["∨.4"]
    , "next_level": ["⇒.7"]
    }


  , "⇒.5":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "unlocks": ["lemmas"]
    , "nodes":
      [ ["a",         [-1,0]]
      , [["⇒",[0,0]], [-1,2.5]]
      , ["b",         [ 1,0]]
      , [["⇒",[1,2]], [ 0,5]]
      ]
    , "hypotheses": [3]
    , "conclusion": 2
    , "text_box" : ["New feature: Lemmas! Wires are clickable now.", "Lemmas"]
    , "map_position": [19.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["∨.5","∧.5"]
    , "next_level": ["⇒.6"]
    }
  , "⇒.6":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-3,0]]
      , ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , ["b",         [ 3,0]]
      , [["⇒",[1,2]], [ 0,3]]
      , [["=",[0,4]], [-1,6]]
      , [["⇒",[5,3]], [ 0,9]]
      ]
    , "hypotheses": []
    , "conclusion": 6
    , "map_position": [21,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["⇒.5"]
    , "next_level": ["⇒.7"]
    }
  , "⇒.7": 
    { "variables": {"a": "Ω", "b": "Ω", "c": "Ω"}
    , "nodes":
      [ ["a",         [-2,0]]
      , ["b",         [ 0,0]]
      , ["c",         [ 2,0]]
      , [["⇒",[0,2]], [ 0,3]]
      , [["⇒",[1,2]], [ 2,3]]
      , [["∨",[0,1]], [-2,3]]
      , [["⇒",[5,2]], [-1,5.5]]
      , [["∧",[3,4]], [ 1,5.5]]
      , [["=",[6,7]], [ 0,8]]
      ]
    , "hypotheses": []
    , "conclusion": 8
    , "map_position": [22.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["⇒.6"]
    , "next_level": ["⇒.8"]
    }
  , "⇒.8":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["⇒",[0,1]], [-1.5,3]]
      , [["⇒",[1,0]], [ 0,3]]
      , [["∧",[2,3]], [-0.75,5]]
      , [["=",[0,1]], [ 1.5,3]]
      , [["=",[4,5]], [ 0,7]]
      ]
    , "hypotheses": []
    , "conclusion": 6
    , "map_position": [24,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["⇒.7"]
    , "next_level": ["¬.3"]
    }


  , "<-asym":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-1,0]]
      , ["b"        , [ 1,0]]
      , [["<",[0,1]], [-1,3]]
      , [["<",[1,0]], [ 1,3]]
      , [["∨",[   ]], [ 3,0]]
      ]
    , "hypotheses": [2,3]
    , "conclusion": 4
    , "map_position": [27,4]
    , "bezier_vector": [1,0]
    , "prereqs": ["⇒.8"]
    , "next_level": ["<.0"]
    }
  , "<-cotrans":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [0,0]]
      , ["b"        , [2,0]]
      , ["c"        , [4,0]]
      , [["<",[0,1]], [1,3]]
      , [["<",[1,2]], [3,3]]
      , [["∨",[3,4]], [2,6]]
      , ["a"        , [-4,0]]
      , ["c"        , [-2,0]]
      , [["<",[6,7]], [-3,3]]
      ]
    , "hypotheses": [8]
    , "conclusion": 5
    , "map_position": [27,6]
    , "bezier_vector": [1,0]
    , "prereqs": ["⇒.8"]
    , "next_level": ["<.1"]
    }
  , "<.0":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [-1,0]]
      , [["<",[0,0]], [-1,3]]
      , [["∨",[   ]], [ 1,0]]
      ]
    , "hypotheses": [1]
    , "conclusion": 2
    , "_comment": "Irreflexivity."
    , "map_position": [28.5,4]
    , "bezier_vector": [1,0]
    , "prereqs": ["<-asym"]
    , "next_level": []
    }
  , "<.1":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a",         [-1.5,0]]
      , ["b",         [ 0  ,0]]
      , ["c",         [ 1.5,0]]
      , [["<",[0,1]], [-1.5,3]]
      , [["<",[1,2]], [ 1.5,3]]
      , [["<",[0,2]], [ 0  ,3]]
      ]
    , "hypotheses": [3,4]
    , "conclusion": 5
    , "_comment": "Transitivity."
    , "map_position": [28.5,6]
    , "bezier_vector": [1,0]
    , "prereqs": ["<-asym", "<-cotrans"]
    , "next_level": []
    }


  , "≠-def":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-1,0]]
      , ["b"        , [ 1,0]]
      , [["<",[0,1]], [-1.5,3]]
      , [["<",[1,0]], [ 0,3]]
      , [["≠",[0,1]], [ 1.5,3]]
      , [["∨",[2,3]], [-0.75,5.5]]
      , [["=",[5,4]], [ 0,8]]
      ]
    , "hypotheses": []
    , "conclusion": 6
    , "map_position": [30,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<.0", "<.1"]
    , "next_level": []
    }
  , "≠.0":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [-1,0]]
      , [["≠",[0,0]], [-1,3]]
      , [["∨",[   ]], [ 1,0]]
      ]
    , "hypotheses": [1]
    , "conclusion": 2
    , "map_position": [31.5,4]
    , "bezier_vector": [1,0]
    , "prereqs": ["≠-def"]
    , "next_level": ["≠.1", "≠.2", "≠-tight"]
    }
  , "≠.1":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-1,0]]
      , ["b"        , [ 1,0]]
      , [["≠",[0,1]], [-1,3]]
      , [["≠",[1,0]], [ 1,3]]
      ]
    , "hypotheses": [2]
    , "conclusion": 3
    , "map_position": [31.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["≠-def"]
    , "next_level": ["≠.2", "≠.0", "≠-tight"]
    }
  , "≠.2":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [0,0]]
      , ["b"        , [2,0]]
      , ["c"        , [4,0]]
      , [["≠",[0,1]], [1,3]]
      , [["≠",[1,2]], [3,3]]
      , [["∨",[3,4]], [2,6]]
      , ["a"        , [-4,0]]
      , ["c"        , [-2,0]]
      , [["≠",[6,7]], [-3,3]]
      ]
    , "hypotheses": [8]
    , "conclusion": 5
    , "map_position": [31.5,6]
    , "bezier_vector": [1,0]
    , "prereqs": ["≠-def"]
    , "next_level": ["≠.0", "≠.1", "≠-tight"]
    }
  , "≠-tight":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-1,0]]
      , ["b"        , [ 1,0]]
      , [["≠",[0,1]], [-1,3]]
      , [["¬",[2  ]], [-1,5]]
      , [["=",[0,1]], [ 1,3]]
      , [["=",[3,4]], [ 0,7.5]]
      ]
    , "hypotheses": []
    , "conclusion": 5
    , "map_position": [33,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["≠.0", "≠.1", "≠.2", "¬.4"]
    , "next_level": ["≠.3"]
    }
  , "≠.3":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-1,0]]
      , ["b"        , [ 1,0]]
      , [["=",[0,1]], [ 0,3]]
      , [["¬",[2  ]], [ 0,5]]
      , [["¬",[3  ]], [ 0,7]]
      ]
    , "hypotheses": [4]
    , "conclusion": 2
    , "map_position": [34.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["≠-tight"]
    , "next_level": []
    }


  , "<+-axiom":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-2,0]]
      , ["b"        , [ 0,0]]
      , ["c"        , [ 2,0]]
      , [["<",[0,1]], [-2,3]]
      , [["+",[0,2]], [ 0,3]]
      , [["+",[1,2]], [ 2,3]]
      , [["<",[4,5]], [ 1,5.5]]
    ]
    , "hypotheses": [3]
    , "conclusion": 6
    , "map_position": [37.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["≠-tight", "−.1"]
    , "next_level": ["<+.0"]
    }
  , "<+.0":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ [["0",[]]   , [-2,0]]
      , ["a"        , [ 0,0]]
      , ["b"        , [ 2,0]]
      , [["<",[0,1]], [-2,3]]
      , [["<",[0,2]], [ 0,3]]
      , [["+",[1,2]], [ 2,3]]
      , [["<",[0,5]], [ 1,5.5]]
    ]
    , "hypotheses": [3,4]
    , "conclusion": 6
    , "map_position": [39,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<+-axiom"]
    , "next_level": ["<+.1"]
    }
  , "<+.1":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-1,0]]
      , ["b"        , [ 1,0]]
      , [["<",[0,1]], [ 0,2.5]]
      , [["-",[0  ]], [-1.25,3]]
      , [["-",[1  ]], [ 1.25,3]]
      , [["<",[4,3]], [ 0,5.5]]
    ]
    , "hypotheses": [2]
    , "conclusion": 5
    , "map_position": [40.5,4.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<+.0"]
    , "next_level": ["<+.2", "<+.3"]
    }
  , "<+.2":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [ 1,0]]
      , ["b"        , [-1,0]]
      , [["<",[0,1]], [ 1,3]]
      , [["−",[1,0]], [-1,3]]
      , [["0",[   ]], [-3,3]]
      , [["<",[4,3]], [-2,5.5]]
      , [["=",[5,2]], [-1,8]]
    ]
    , "hypotheses": []
    , "conclusion": 6
    , "map_position": [40.5,5.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<+.0"]
    , "next_level": ["<+.1", "<+.3"]
    }
  , "<+.3":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-2,0]]
      , ["b"        , [ 0,0]]
      , ["c"        , [ 2,0]]
      , [["≤",[0,1]], [-2,3]]
      , [["+",[0,2]], [ 0,3]]
      , [["+",[1,2]], [ 2,3]]
      , [["≤",[4,5]], [ 1,5.5]]
    ]
    , "hypotheses": [3]
    , "conclusion": 6
    , "map_position": [42,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<+.1", "<+.2", "≤.1"]
    , "next_level": []
    }


  , "0<1":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ [["0",[]]   , [-1,0]]
      , [["1",[]]   , [ 1,0]]
      , [["<",[0,1]], [ 0,3]]
    ]
    , "hypotheses": []
    , "conclusion": 2
    , "map_position": [45,4.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<+.3"]
    , "next_level": []
    }
  , "<×-axiom":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ [["0",[]]   , [-2,0]]
      , ["a"        , [ 0,0]]
      , ["b"        , [ 2,0]]
      , [["<",[0,1]], [-2,3]]
      , [["<",[0,2]], [ 0,3]]
      , [["×",[1,2]], [ 2,3]]
      , [["<",[0,5]], [ 1,5.5]]
    ]
    , "hypotheses": [3,4]
    , "conclusion": 6
    , "map_position": [45,5.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<+.3", "×.6"]
    , "next_level": ["<×.0"]
    }
  , "<×.0":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-2,0]]
      , ["b"        , [ 0,0]]
      , ["c"        , [ 2,0]]
      , [["<",[0,1]], [-2,3]]
      , [["×",[0,2]], [ 0,3]]
      , [["×",[1,2]], [ 2,3]]
      , [["<",[4,5]], [ 1,5.5]]
      , [["0",[   ]], [ 5,0]]
      , ["c"        , [ 7,0]]
      , [["<",[7,8]], [ 6,3]]
    ]
    , "hypotheses": [3,9]
    , "conclusion": 6
    , "map_position": [46.5,5.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<×-axiom"]
    , "next_level": ["<×.1"]
    }
  , "<×.1":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ [["0",[]]   , [-2,0]]
      , ["a"        , [ 0,0]]
      , ["b"        , [ 2,0]]
      , [["≤",[0,1]], [-2,3]]
      , [["≤",[0,2]], [ 0,3]]
      , [["×",[1,2]], [ 2,3]]
      , [["≤",[0,5]], [ 1,5.5]]
    ]
    , "hypotheses": [3,4]
    , "conclusion": 6
    , "map_position": [48,5.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<×.0", "trichotomy"]
    , "next_level": ["<×.2"]
    }
  , "<×.2":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [ 0,0]]
      , [["×",[0,0]], [ 0,3]]
      , [["0",[   ]], [-2,3]]
      , [["≤",[2,1]], [-1,5.5]]
    ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [49.5,5.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<×.1"]
    , "next_level": ["⅟.0"]
    }


  , "⅟.0":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-2,0]]
      , ["b"        , [ 0,0]]
      , [["0",[   ]], [ 2,0]]
      , [["×",[0,1]], [-1,3]]
      , [["≠",[1,2]], [ 1,3]]
      , [["1",[   ]], [-3,3]]
      , [["=",[5,3]], [-2,5.5]]
      ]
    , "hypotheses": [6]
    , "conclusion": 4
    , "map_position": [54,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["<×.2", "0<1"]
    , "next_level": ["×-inv"]
    }
  , "×-inv":
    { "axiom": true
    , "variables": {"a": "ℝ"}
    , "nodes":
      [ [["0",[   ]], [ 2  ,0]]
      , ["a"        , [ 0  ,0]]
      , [["≠",[1,0]], [ 1  ,2.5]]
      , [["⅟",[1  ]], [-1  ,3]]
      , [["×",[1,3]], [ 0.5,4.5]]
      , [["1",[   ]], [ 2.5,4.5]]
      , [["=",[4,5]], [ 1.5,7]]
      ]
    , "hypotheses": [2]
    , "conclusion": 6
    , "map_position": [55.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["⅟.0"]
    , "next_level": ["⅟.1", "⅟.2"]
    }
  , "⅟.1":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a",         [-1.5,0]]
      , ["b",         [ 0  ,0]]
      , ["c",         [ 1.5,0]]
      , [["=",[0,1]], [-1.5,3]]
      , [["×",[0,2]], [ 0  ,3]]
      , [["×",[1,2]], [ 1.5,3]]
      , [["=",[4,5]], [ 0.75,5.5]]
      , [["0",[   ]], [ 3,0]]
      , [["≠",[2,7]], [ 3,3]]
      ]
    , "hypotheses": [6,8]
    , "conclusion": 3
    , "map_position": [57,5.75]
    , "bezier_vector": [1,0]
    , "prereqs": ["×-inv"]
    , "next_level": ["IntegralDomain", "⅟.3"]
    }
  , "IntegralDomain":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-2,0]]
      , ["b"        , [ 0,0]]
      , [["0",[   ]], [ 2,0]]
      , [["×",[0,1]], [-2,3]]
      , [["=",[0,2]], [ 0,3]]
      , [["=",[1,2]], [ 2,3]]
      , [["=",[3,2]], [-1,5.5]]
      , [["∨",[4,5]], [ 1,5.5]]
    ]
    , "hypotheses": [6]
    , "conclusion": 7
    , "map_position": [58.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["⅟.1"]
    , "next_level": ["¬IntegralDomain"]
    }
  , "¬IntegralDomain":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a"        , [-2,0]]
      , ["b"        , [ 0,0]]
      , [["0",[   ]], [ 2,0]]
      , [["×",[0,1]], [-2,3]]
      , [["≠",[0,2]], [ 0,3]]
      , [["≠",[1,2]], [ 2,3]]
      , [["≠",[3,2]], [-1,5.5]]
    ]
    , "hypotheses": [4,5]
    , "conclusion": 6
    , "map_position": [60,4.25]
    , "bezier_vector": [1,0]
    , "prereqs": ["IntegralDomain"]
    , "next_level": ["÷-def"]
    }
  , "⅟.2":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ [["0",[   ]], [-1,0]]
      , ["a"        , [ 1,0]]
      , [["<",[0,1]], [ 0,2.5]]
      , [["0",[   ]], [-1.25,3]]
      , [["⅟",[1  ]], [ 1.25,3]]
      , [["<",[3,4]], [ 0,5.5]]
    ]
    , "hypotheses": [2]
    , "conclusion": 5
    , "map_position": [57,4.25]
    , "bezier_vector": [1,0]
    , "prereqs": ["×-inv"]
    , "next_level": []
    }
  , "⅟.3":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a"        , [-2,0]]
      , ["b"        , [ 0,0]]
      , [["⅟",[0  ]], [-2,2.5]]
      , [["×",[0,1]], [ 0,3]]
      , [["1",[   ]], [ 2,2.5]]
      , [["=",[2,1]], [-2,5]]
      , [["=",[3,4]], [ 1,5]]
    ]
    , "hypotheses": [6]
    , "conclusion": 5
    , "map_position": [58.5,6.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["⅟.1"]
    , "next_level": ["⅟.4","⅟.5"]
    }
  , "⅟.4":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["a",         [ 1,0]]
      , [["⅟",[1  ]], [ 1,2.5]]
      , [["⅟",[2  ]], [ 1,4.5]]
      , [["=",[0,3]], [ 0,6.5]]
      , [["0",[   ]], [ 3,0]]
      , [["≠",[1,5]], [ 3,2.5]]
      ]
    , "hypotheses": [6]
    , "conclusion": 4
    , "map_position": [60,7.25]
    , "bezier_vector": [1,0]
    , "prereqs": ["⅟.3"]
    , "next_level": ["÷-def"]
    }
  , "⅟.5":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [ 1,0]]
      , [["-",[0  ]], [-1,2.5]]
      , [["⅟",[1  ]], [-1,4.5]]
      , [["⅟",[0  ]], [ 1,2.5]]
      , [["-",[3  ]], [ 1,4.5]]
      , [["=",[2,4]], [ 0,6.5]]
      , [["0",[   ]], [ 3,0]]
      , [["≠",[0,6]], [ 3,2.5]]
      ]
    , "hypotheses": [7]
    , "conclusion": 5
    , "map_position": [60,5.75]
    , "bezier_vector": [1,0]
    , "prereqs": ["⅟.3"]
    , "next_level": ["÷-def"]
    }


  , "÷-def":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["⅟",[1  ]], [ 1,2.5]]
      , [["×",[0,2]], [ 1,5]]
      , [["÷",[0,1]], [-1,3.5]]
      , [["=",[4,3]], [ 0,7]]
      ]
    , "hypotheses": []
    , "conclusion": 5
    , "map_position": [64.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["⅟.5","⅟.4","¬IntegralDomain"]
    , "next_level": ["1÷-correct"]
    }
  , "1÷-correct":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ [["1",[   ]], [-1,0]]
      , ["a",         [ 1,0]]
      , [["÷",[0,1]], [ 0,3]]
      , [["⅟",[1  ]], [ 2,3]]
      , [["=",[2,3]], [ 1,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [66,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["÷-def"]
    , "next_level": ["÷.0","÷.2","÷.1"]
    }
  , "÷.0":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["÷",[0,0]], [ 0,3]]
      , [["1",[   ]], [-2,3]]
      , [["=",[2,1]], [-1,5.5]]
      , [["0",[   ]], [ 2,0]]
      , [["≠",[0,4]], [ 2,3]]
      ]
    , "hypotheses": [5]
    , "conclusion": 3
    , "map_position": [67.5,4]
    , "bezier_vector": [1,0]
    , "prereqs": ["1÷-correct"]
    , "next_level": ["÷.2","÷.1","÷.3"]
    }
  , "÷.1":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ", "d": "ℝ"}
    , "nodes":
      [ ["a",         [-3,0]]
      , ["b",         [-1,0]]
      , ["c",         [ 1,0]]
      , ["d",         [ 3,0]]
      , [["÷",[0,1]], [-3,3]]
      , [["×",[0,2]], [-1,3]]
      , [["×",[1,3]], [ 1,3]]
      , [["÷",[2,3]], [ 3,3]]
      , [["×",[4,7]], [-1,6]]
      , [["÷",[5,6]], [ 1,6]]
      , [["=",[8,9]], [ 0,8.5]]
      , ["b",           [ 6,0]]
      , [["0",[     ]], [ 8,0]]
      , [["≠",[11,12]], [ 7,2.5]]
      , ["d",           [ 6,6]]
      , [["0",[     ]], [ 8,6]]
      , [["≠",[14,15]], [ 7,8.5]]
      ]
    , "hypotheses": [13,16]
    , "conclusion": 10
    , "map_position": [67.5,6]
    , "bezier_vector": [1,0]
    , "prereqs": ["1÷-correct"]
    , "next_level": ["÷.0","÷.2","÷.3"]
    }
  , "÷.2":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a",         [-2,0]]
      , ["b",         [ 0,0]]
      , ["c",         [ 2,0]]
      , [["+",[0,1]], [-2,3]]
      , [["÷",[0,2]], [ 0,3]]
      , [["÷",[1,2]], [ 2,3]]
      , [["÷",[3,2]], [-1,6]]
      , [["+",[4,5]], [ 1,6]]
      , [["=",[6,7]], [ 0,8.5]]
      , ["c",          [ 5,0]]
      , [["0",[    ]], [ 7,0]]
      , [["≠",[9,10]], [ 6,2.5]]
      ]
    , "hypotheses": [11]
    , "conclusion": 8
    , "map_position": [67.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["1÷-correct"]
    , "next_level": ["÷.1","÷.0","÷.3"]
    }
  , "÷.3":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ", "d": "ℝ"}
    , "nodes":
      [ ["a",         [-3.5,-0.5]]
      , ["b",         [-1.5,-0.5]]
      , ["c",         [ 1.5,-0.5]]
      , ["d",         [ 3.5,-0.5]]
      , [["÷",[0,1]], [-4,3]]
      , [["×",[0,3]], [-2,3]]
      , [["×",[1,2]], [ 0,3]]
      , [["×",[1,3]], [ 2,3]]
      , [["÷",[2,3]], [ 4,3]]

      , [["+",[4,8]], [-2,7]]
      , [["+",[5,6]], [ -1,5]]
      , [["÷",[10,7]], [ 0,7]]
      , [["=",[9,11]], [-1,9.5]]
      , ["b",           [ 6,0]]
      , [["0",[     ]], [ 8,0]]
      , [["≠",[13,14]], [ 7,2.5]]
      , ["d",           [ 6,6]]
      , [["0",[     ]], [ 8,6]]
      , [["≠",[16,17]], [ 7,8.5]]
      ]
    , "hypotheses": [15,18]
    , "conclusion": 12
    , "map_position": [69,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["÷.0","÷.2","÷.1"]
    , "next_level": ["÷.4"]
    }
  , "÷.4":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["+",[0,1]], [-1,3]]
      , [["×",[0,1]], [ 1,3]]
      , [["÷",[2,3]], [ 0,6]]
      , ["a",         [-6,0]]
      , ["b",         [-4,0]]
      , [["⅟",[5  ]], [-6,3]]
      , [["⅟",[6  ]], [-4,3]]
      , [["+",[7,8]], [-5,6]]
      , [["=",[9,4]], [-2.5,8.5]]
      , ["a",           [ 6,0]]
      , [["0",[     ]], [ 8,0]]
      , [["≠",[11,12]], [ 7,2.5]]
      , ["b",           [ 6,6]]
      , [["0",[     ]], [ 8,6]]
      , [["≠",[14,15]], [ 7,8.5]]
      ]
    , "hypotheses": [13,16]
    , "conclusion": 10
    , "map_position": [70.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["÷.3"]
    , "next_level": ["÷.5"]
    }
  , "÷.5":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [ 1,0]]
      , ["b",         [-1,0]]
      , [["−",[1,0]], [-1,3]]
      , [["×",[1,0]], [ 1,3]]
      , [["÷",[2,3]], [ 0,6]]
      , ["a",         [-6,0]]
      , ["b",         [-4,0]]
      , [["⅟",[5  ]], [-6,3]]
      , [["⅟",[6  ]], [-4,3]]
      , [["−",[7,8]], [-5,6]]
      , [["=",[9,4]], [-2.5,8.5]]
      , ["a",           [ 6,0]]
      , [["0",[     ]], [ 8,0]]
      , [["≠",[11,12]], [ 7,2.5]]
      , ["b",           [ 6,6]]
      , [["0",[     ]], [ 8,6]]
      , [["≠",[14,15]], [ 7,8.5]]
      ]
    , "hypotheses": [13,16]
    , "conclusion": 10
    , "map_position": [72,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["÷.4"]
    , "next_level": ["^+","^×"]
    }


  , "^+":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a",         [-2,0]]
      , ["b",         [ 0,0]]
      , ["c",         [ 2,0]]
      , [["^",[0,1]], [-2,3]]
      , [["^",[0,2]], [ 0,3]]
      , [["+",[1,2]], [ 2,3]]
      , [["×",[3,4]], [-1,5.5]]
      , [["^",[0,5]], [ 1,5.5]]
      , [["=",[6,7]], [ 0,8]]
      , [["0",[   ]], [-4,0]]
      , [["<",[9,0]], [-4,3]]
      ]
    , "hypotheses": [10]
    , "conclusion": 8
    , "map_position": [76.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["÷.4"]
    , "next_level": ["^>0"]
    }
  , "^>0":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , ["b",         [ 2,0]]
      , [["^",[0,1]], [ 1,3]]
      , [["0",[   ]], [-2,0]]
      , [["<",[3,0]], [-1,3]]
      , [["<",[3,2]], [-1,5.5]]
      ]
    , "hypotheses": [4]
    , "conclusion": 5
    , "map_position": [78,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["^+"]
    , "next_level": ["^0"]
    }
  , "^0":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["0",[   ]], [ 2,0]]
      , [["^",[0,1]], [ 1,3]]
      , [["1",[   ]], [ 4,0]]
      , [["=",[2,3]], [ 2,5.5]]
      , [["0",[   ]], [-2,0]]
      , [["<",[5,0]], [-1,3]]
      ]
    , "hypotheses": [6]
    , "conclusion": 4
    , "map_position": [79.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["^>0"]
    , "next_level": ["^1", "^×"]
    }
  , "^-":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , ["b",         [ 2,0]]
      , [["-",[1  ]], [ 2,3]]
      , [["^",[0,1]], [ 0,3]]
      , [["⅟",[3  ]], [ 0,5.5]]
      , [["^",[0,2]], [ 2,5.5]]
      , [["=",[4,5]], [ 1,8]]
      , [["0",[   ]], [-2,0]]
      , [["<",[7,0]], [-2,3]]
      ]
    , "hypotheses": [8]
    , "conclusion": 6
    , "map_position": [81,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["^0"]
    , "next_level": ["^×","^1"]
    }
  , "^1":
    { "axiom": true
    , "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["1",[   ]], [ 2,0]]
      , [["^",[0,1]], [ 1,3]]
      , [["=",[0,2]], [ 0,5.5]]
      , [["0",[   ]], [-2,0]]
      , [["<",[4,0]], [-1,3]]
      ]
    , "hypotheses": [5]
    , "conclusion": 3
    , "map_position": [82.5,4.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["^-"]
    , "next_level": ["^×","^-1"]
    }
  , "^×":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a",         [-1.5,0]]
      , ["b",         [ 0  ,0]]
      , ["c",         [ 1.5,0]]
      , [["^",[0,1]], [-1,3.5]]
      , [["×",[1,2]], [ 1,3.5]]
      , [["^",[3,2]], [-1,5.5]]
      , [["^",[0,4]], [ 1,5.5]]
      , [["=",[5,6]], [ 0,7.5]]
      , [["0",[   ]], [-3.5,0]]
      , [["<",[8,0]], [-2.5,3.5]]
      ]
    , "hypotheses": [9]
    , "conclusion": 7
    , "map_position": [82.5,5.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["^-"]
    , "next_level": ["^1","^-1"]
    }
  , "^-1":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["-1",[  ]], [ 2,0]]
      , [["^",[0,1]], [ 2,3]]
      , [["⅟",[0  ]], [ 0,3]]
      , [["=",[3,2]], [ 1,5.5]]
      , [["0",[   ]], [-2,0]]
      , [["<",[5,0]], [-2,3]]
      ]
    , "hypotheses": [6]
    , "conclusion": 4
    , "map_position": [84,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["^×","^1"]
    , "next_level": ["√-def"]
    }


  , "√-def":
    { "axiom": true
    , "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["0",[   ]], [-2,0]]
      , [["≤",[1,0]], [-1,3]]
      , [["√",[0  ]], [ 1,3]]
      , [["0",[   ]], [-3,3]]
      , [["≤",[4,3]], [-1,6]]
      , [["×",[3,3]], [ 1,6]]
      , ["a",         [ 3,6]]
      , [["=",[6,7]], [ 2,8]]
      , [["∧",[5,8]], [ 0.5,10]]
      ]
    , "hypotheses": [2]
    , "conclusion": 9
    , "map_position": [87,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["^-1"]
    , "next_level": ["√0"]
  }
  , "√0":
    { "variables": {}
    , "nodes":
      [ [["0",[   ]], [-1,0]]
      , [["√",[0  ]], [-1,3]]
      , [["0",[   ]], [ 1,0]]
      , [["=",[1,2]], [ 0,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [88.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["√-def", "2^2=4"]
    , "next_level": ["depressed-monic-quadratic"]
    }
  , "depressed-monic-quadratic":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a"        , [-2,0]]
      , [["×",[0,0]], [-2,3]]
      , [["√",[1  ]], [-2,5]]
      , ["a"        , [ 0,5]]
      , ["a"        , [ 2,3]]
      , [["-",[4  ]], [ 2,5]]
      , [["=",[2,3]], [-1,8]]
      , [["=",[2,5]], [ 1,8]]
      , [["∨",[6,7]], [ 0,10.5]]
    ]
    , "hypotheses": []
    , "conclusion": 8
    , "map_position": [90,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["√0"]
    , "next_level": ["√-power"]
    }
  , "√-power":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["0",[   ]], [-2,0]]
      , [["<",[1,0]], [-2,3]]
      , [["√",[0  ]], [ 0,3]]
      , [["½",[   ]], [ 2,0]]
      , [["^",[0,4]], [ 2,3]]
      , [["=",[3,5]], [ 1,5.5]]
      ]
    , "hypotheses": [2]
    , "conclusion": 6
    , "map_position": [91.5,5.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["depressed-monic-quadratic"]
    , "next_level": []
    }
  , "√×√":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ [["0",[   ]], [-3,0]]
      , ["a",         [ 0,0]]
      , ["b",         [ 3,0]]
      , [["≤",[0,1]], [-4,3]]
      , [["≤",[0,2]], [-2,3]]
      , [["√",[1  ]], [ 0,3]]
      , [["×",[1,2]], [ 2,3]]
      , [["√",[2  ]], [ 4,3]]
      , [["×",[5,7]], [ 1,6]]
      , [["√",[6  ]], [ 3,6]]
      , [["=",[8,9]], [ 2,8.5]]
      ]
    , "hypotheses": [3,4]
    , "conclusion": 10
    , "map_position": [91.5,4.5]
    , "bezier_vector": [1,0]
    , "prereqs": ["depressed-monic-quadratic"]
    , "next_level": []
    }


  , "Quadratic.0":
    { "variables": {"z": "ℝ", "a": "ℝ"}
    , "nodes":
      [ ["z"        , [-2,3]]
      , ["a"        , [ 1,0]]
      , [["2",[   ]], [ 3,0]]
      , [["÷",[1,2]], [ 2,3]]
      , [["×",[0,0]], [-3,6]]
      , [["×",[0,1]], [-1,6]]
      , [["×",[3,3]], [ 1,6]]
      , [["+",[4,5]], [-2,8]]
      , [["+",[7,6]], [-1,10]]
      , [["+",[0,3]], [ 3,6]]
      , [["×",[9,9]], [ 3,10]]
      , [["=",[8,10]], [1,12.5]]
    ]
    , "hypotheses": []
    , "conclusion": 11
    , "map_position": [96,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["√×√", "√-power"]
    , "next_level": ["Quadratic.1"]
    , "text_box": ["I wanted to use the variable x, but it looks too much like ×."]
    }
  , "Quadratic.1":
    { "variables": {"z": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["z"        , [-2,0]]
      , ["b"        , [ 0,0]]
      , [["×",[0,0]], [-2,3]]
      , [["×",[0,1]], [ 0,3]]
      , ["c"        , [ 2,3]]
      , [["+",[2,3]], [-1,5]]
      , [["+",[5,4]], [ 0,7]]
      , [["0",[   ]], [ 2,7]]
      , [["=",[6,7]], [ 1,9.5]]
      
      , ["b"        , [ 8,0]]
      , [["2",[   ]], [10,0]]
      , [["÷",[9,10]],[ 9,2]]
      , [["×",[11,11]],[10,4]]
      , ["c"        , [13,0]]
      , [["−",[12,13]],[11,6]]
      , [["√",[14 ]], [10,8]]
      , [["-",[11 ]], [ 8,8]]
      , [["+",[16,15]], [ 8 ,11]]
      , [["−",[16,15]], [ 10,11]]
      , ["z"          , [ 6 ,11]]
      , [["=",[19,17]], [ 7,14]]
      , [["=",[19,18]], [ 9,14]]
      , [["∨",[20,21]], [ 8,16.5]]
    ]
    , "hypotheses": [8]
    , "conclusion": 22
    , "map_position": [97.5,5]
    , "bezier_vector": [1,0]
    , "prereqs": ["Quadratic.0"]
    , "next_level": []
    , "text_box": ["A variation of the quadratic formula."]
    }








    , "¬-def":
    { "axiom": true
    , "variables": {"a": "Ω"}
    , "nodes":
      [ ["a",         [-0.25,0]]
      , [["∨",[]   ], [ 2,0]]
      , [["¬",[0]  ], [-1,3]]
      , [["⇒",[0,1]], [ 1,3]]
      , [["=",[2,3]], [ 0,6]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [24,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["∨.5"]
    , "next_level": ["¬.0"]
    }
  , "¬.0":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["¬",[0]  ], [-1,3]]
      ]
    , "hypotheses": [0,2]
    , "conclusion": 1
    , "map_position": [25.5,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["¬-def"]
    , "next_level": ["¬.1"]
    }
  , "¬.1":
    { "variables": {"a": "Ω"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["¬",[0]  ], [ 0,2]]
      , [["¬",[1]  ], [ 0,4]]
      ]
    , "hypotheses": [0]
    , "conclusion": 2
    , "map_position": [27,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["¬.0"]
    , "next_level": ["¬.2"]
    }
  , "¬.2":
    { "variables": {"a": "Ω"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["¬",[0]  ], [ 0,2]]
      , [["¬",[1]  ], [ 0,4]]
      , [["¬",[2]  ], [ 0,6]]
      ]
    , "hypotheses": [3]
    , "conclusion": 1
    , "map_position": [28.5,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["¬.1"]
    , "next_level": ["¬.3"]
    }
  , "¬.3":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1.25,0]]
      , ["b",         [ 1.25,0]]
      , [["¬",[0]  ], [-2,3]]
      , [["¬",[1]  ], [ 2,3]]
      , [["∨",[0,1]], [ 0,3]]
      , [["∧",[2,3]], [-1,6]]
      , [["¬",[4]  ], [ 1,6]]
      , [["=",[5,6]], [ 0,9]]
      ]
    , "hypotheses": []
    , "conclusion": 7
    , "map_position": [30,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["¬.2", "⇒.8"]
    , "next_level": ["¬.4"]
    }
  , "¬.4":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1.25,0]]
      , ["b",         [ 1.25,0]]
      , [["¬",[0]  ], [-2,3]]
      , [["¬",[1]  ], [ 2,3]]
      , [["∧",[0,1]], [ 0,3]]
      , [["∨",[2,3]], [-1,6]]
      , [["¬",[4]  ], [ 1,6]]
      ]
    , "hypotheses": [5]
    , "conclusion": 6
    , "map_position": [31.5,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["¬.3"]
    , "next_level": []
    }


  , "≤-def":
    { "axiom": true
    , "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["≤",[0,1]], [-1,3]]
      , [["<",[1,0]], [ 1,3]]
      , [["¬",[3]  ], [ 1,5]]
      , [["=",[2,4]], [ 0,7]]
      ]
    , "hypotheses": []
    , "conclusion": 5
    , "map_position": [36,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["≠.3", "¬.4"]
    , "next_level": ["≤.1"]
    }
  , "≤.0":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["<",[0,1]], [-1,3]]
      , [["≤",[1,0]], [ 1,3]]
      , [["∨",[   ]], [ 3,0]]
      ]
    , "hypotheses": [2,3]
    , "conclusion": 4
    , "map_position": [37.5,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["≤-def"]
    , "next_level": ["≤-refl"]
    }
  , "≤-refl":
    { "variables": {"a": "ℝ"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["≤",[0,0]], [ 0,3]]
      ]
    , "hypotheses": []
    , "conclusion": 1
    , "map_position": [39,9]
    , "bezier_vector": [1,0]
    , "prereqs": ["≤.0"]
    , "next_level": ["≤-sym","≤-trans","≤.1"]
    }
  , "≤-sym":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["≤",[0,1]], [-1.5,3]]
      , [["≤",[1,0]], [ 0,3]]
      , [["=",[0,1]], [ 1.5,3]]
      ]
    , "hypotheses": [2,3]
    , "conclusion": 4
    , "map_position": [39,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["≤.0"]
    , "next_level": ["≤-trans","≤-refl","≤.1"]
    }
  , "≤-trans":
    { "variables": {"a": "ℝ", "b": "ℝ", "c": "ℝ"}
    , "nodes":
      [ ["a",         [-1.5,0]]
      , ["b",         [ 0  ,0]]
      , ["c",         [ 1.5,0]]
      , [["≤",[0,1]], [-1.5,3]]
      , [["≤",[0,2]], [ 0  ,3]]
      , [["≤",[1,2]], [ 1.5,3]]
      ]
    , "hypotheses": [3,5]
    , "conclusion": 4
    , "map_position": [39,11]
    , "bezier_vector": [1,0]
    , "prereqs": ["≤.0"]
    , "next_level": ["≤-refl","≤-sym","≤.1"]
    }
  , "≤.1":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["<",[0,1]], [-1,3]]
      , [["≤",[0,1]], [ 1,3]]
      ]
    , "hypotheses": [2]
    , "conclusion": 3
    , "map_position": [40.5,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["≤-refl","≤-sym","≤-trans"]
    , "next_level": ["≤.2"]
    }
  , "≤.2":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["<",[0,1]], [-1,3]]
      , [["≤",[1,0]], [ 1,3]]
      , [["∨",[2,3]], [ 0,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [42,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["≤.1","Classical.4"]
    , "next_level": ["≤.3"]
    }
  , "≤.3":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["≤",[0,1]], [-1.5,3]]
      , [["<",[0,1]], [ 0,3]]
      , [["=",[0,1]], [ 1.5,3]]
      , [["∨",[3,4]], [ 0.75,5.5]]
      ]
    , "hypotheses": [2]
    , "conclusion": 5
    , "map_position": [43.5,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["≤.2"]
    , "next_level": ["trichotomy"]
    }
  , "trichotomy":
    { "variables": {"a": "ℝ", "b": "ℝ"}
    , "nodes":
      [ ["a",         [-1  ,0]]
      , ["b",         [ 1  ,0]]
      , [["<",[0,1]], [-1.5,3]]
      , [["<",[1,0]], [ 0  ,3]]
      , [["=",[0,1]], [ 1.5,3]]
      , [["∨",[2,3,4]], [ 0,5.5]]
      ]
    , "hypotheses": []
    , "conclusion": 5
    , "map_position": [45,10]
    , "bezier_vector": [1,0]
    , "prereqs": ["≤.3"]
    , "next_level": []
    }








  , "¬¬Elim":
    { "axiom": true
    , "variables": {"a": "Ω"}
    , "nodes":
      [ ["a",         [ 0,0]]
      , [["¬",[0]  ], [ 0,2]]
      , [["¬",[1]  ], [ 0,4]]
      ]
    , "hypotheses": [2]
    , "conclusion": 0
    , "map_position": [33,15]
    , "bezier_vector": [1,0]
    , "prereqs": ["¬.4"]
    , "next_level": ["Classical.0"]
    , "text_box": ["Surprisingly, this can't be proven from the other rules!", "Classical"]
    }
  , "Classical.0":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [ 1,0]]
      , [["¬",[0]  ], [ 1,2]]
      , [["¬",[1]  ], [ 1,4]]
      , ["a",         [-1,4]]
      , [["=",[3,2]], [ 0,7]]
      ]
    , "hypotheses": []
    , "conclusion": 4
    , "map_position": [34.5,15]
    , "bezier_vector": [0,0]
    , "prereqs": ["¬¬Elim"]
    , "next_level": ["Classical.1"]
    }
  , "Classical.1":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1.25,0]]
      , ["b",         [ 1.25,0]]
      , [["¬",[0]  ], [-2,3]]
      , [["¬",[1]  ], [ 2,3]]
      , [["∧",[0,1]], [ 0,3]]
      , [["∨",[2,3]], [-1,6]]
      , [["¬",[4]  ], [ 1,6]]
      , [["=",[5,6]], [ 0,9]]
      ]
    , "hypotheses": []
    , "conclusion": 7
    , "map_position": [36,15]
    , "bezier_vector": [1,0]
    , "prereqs": ["Classical.0"]
    , "next_level": ["Classical.2"]
    }
  , "Classical.2":
    { "variables": {"a": "Ω"}
    , "nodes":
      [ ["a",         [ 1,0]]
      , [["¬",[0]  ], [ 1,2]]
      , ["a",         [-1,0]]
      , [["∨",[2,1]], [ 0,5]]
      ]
    , "hypotheses": []
    , "conclusion": 3
    , "map_position": [37.5,15]
    , "bezier_vector": [0,0]
    , "prereqs": ["Classical.1"]
    , "next_level": ["Classical.3"]
    , "text_box": ["The Law of Excluded Middle."]
    }
  , "Classical.3":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["⇒",[0,1]], [ 1,3]]
      , [["¬",[0]  ], [-1,2.5]]
      , [["∨",[3,1]], [-1,5]]
      , [["=",[4,2]], [ 0,8]]
      ]
    , "hypotheses": []
    , "conclusion": 5
    , "map_position": [39,15]
    , "bezier_vector": [0,0]
    , "prereqs": ["Classical.2"]
    , "next_level": ["Classical.4"]
    }
  , "Classical.4":
    { "variables": {"a": "Ω", "b": "Ω"}
    , "nodes":
      [ ["a",         [-1,0]]
      , ["b",         [ 1,0]]
      , [["⇒",[0,1]], [-1,3]]
      , [["¬",[2]  ], [-1,5]]
      , [["¬",[1]  ], [ 1,2]]
      , [["∧",[0,4]], [ 1,5]]
      , [["=",[3,5]], [ 0,8]]
      ]
    , "hypotheses": []
    , "conclusion": 6
    , "map_position": [40.5,15]
    , "bezier_vector": [1,0]
    , "prereqs": ["Classical.3"]
    , "next_level": []
    }


  }
}