data/peano-arithmetic-addition-multiplication.mmd

graph TD
    A5["A5\nInduction"]
    DefAdd["DefAdd\nDefinition of +"]
    T5["T5\nAssociativity of +"]
    T6["T6\nLeft identity"]
    T7["T7\nSuccessor and add"]
    T8["T8\nCommutativity of +"]
    T9["T9\nCancellation for +"]
    DefMul["DefMul\nDefinition of ·"]
    T10["T10\nMul well-defined"]
    T11["T11\nZero times"]
    T12["T12\nZero from left"]
    T13["T13\nSuccessor and mul"]
    T14["T14\nCommutativity of ·"]
    T15["T15\nAssociativity of ·"]
    T16["T16\nDistributivity"]
    T17["T17\nDistributivity (right)"]
    A5 --> DefAdd
    DefAdd --> T5
    A5 --> T5
    DefAdd --> T6
    A5 --> T6
    DefAdd --> T7
    T6 --> T7
    A5 --> T7
    DefAdd --> T8
    T5 --> T8
    T6 --> T8
    T7 --> T8
    A5 --> T8
    DefAdd --> T9
    T8 --> T9
    A5 --> T9
    DefAdd --> DefMul
    A5 --> DefMul
    DefMul --> T10
    A5 --> T10
    DefMul --> T11
    DefMul --> T12
    T6 --> T12
    A5 --> T12
    DefMul --> T13
    T8 --> T13
    A5 --> T13
    DefMul --> T14
    T12 --> T14
    T13 --> T14
    A5 --> T14
    DefMul --> T15
    T5 --> T15
    T8 --> T15
    A5 --> T15
    DefMul --> T16
    T5 --> T16
    T8 --> T16
    T15 --> T16
    A5 --> T16
    T16 --> T17
    T8 --> T17
    classDef axiom fill:#e74c3c,color:#fff,stroke:#c0392b
    classDef definition fill:#3498db,color:#fff,stroke:#2980b9
    classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085
    class A5 axiom
    class DefAdd,DefMul definition
    class T5,T6,T7,T8,T9,T10,T11,T12,T13,T14,T15,T16,T17 theorem