[
    {
        "problem": "For any natural number n, 0 + n = n.",
        "level": "SF foundations level 1",
        "type": "Logical Foundations",
        "solution": [
            "Consider some natural number n. We want to show 0 + n = n. ",
            "By using definition of addition on both sides, LHS and RHS are now equal, done." 
        ]
    },
    {
        "problem": "For any natural number n, n + 0 = n.",
        "level": "SF foundations level 1",
        "type": "Logical Foundations",
        "solution": [
            "Consider some natural number n. The proof will be by induction. ",
            "The base case n=0, so we have 0 + 0 = 0, which holds by the definition of addion. ",
            "Consider the inductive case, so we want to show (k + 1) + 0 = (k + 1) for any k < n assuming the IH holds for such k (IH: k + 0 = k). ",
            "By the IH we have (k + 1) + 0 = (k + 1). ",
            "By def of addition we have (k + 0) + 1 = (k + 1). ", 
            "By the induction hypothesis (IH) we have k + 0 = k. ", 
            "LHS and RHS are equal so proof is complete."
        ] 
    }
]