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Proof step 44 from proof copy2append0 (previous step)

# ar1 = (n2 + n1) + n0, n3 < n2 + n1, n3 < # ar0, n0 < # ar1, n3 < n1, n3 < n1 + n2, n1 + n0 ≤ # ar1, n1 ≤ # ar0, n0 + n1 ≤ # ar1 ⊦ ar0[n3] = (subarray(ar0, 0, n1) + subarray(ar1, n0 + n1, n2))[n3]

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: append-at1 seq: ⊦ n < # ar0 → (ar0 + ar1)[n] = ar0[n] subst: [n, ar1, ar0] → [n3, subarray(ar1, n0 + n1, n2), subarray(ar0, 0, n1)] flags: <all paths>

Additional information

Used simplifier rules: append-at1: ⊦ n < # ar0 → (ar0 + ar1)[n] = ar0[n];%append-a: ⊦ (ar + ar0) + ar1 = ar + ar0 + ar1;% ⊦ n0 < n → subarray(ar, m, n)[n0] = ar[m + n0]; (s)% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ # subarray(ar, m, n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%# ar = n ⊦ subarray(ar, 0, n) = ar; (s) Used terms: n3, subarray(ar1, n0 + n1, n2), subarray(ar0, 0, n1)