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Proof step 43 from proof datasize-append-empty-h (previous step)

m0 + 1 = n0 + # buf0, datasize(buf1, n0) = 0, m0 < # buf0 + # buf1, # buf0 ≤ m0 + 1, n0 ≤ # buf1, ∀ buf0, buf1. m0 ≤ # buf0 + # buf1 ∧ (# buf0 ≤ m0 → datasize(buf1, m0 - # buf0) = 0) → datasize(buf0 + buf1, m0) = datasize(buf0, min(m0, # buf0)) ⊦ m0 + 1 = datasize(buf0, # buf0), (buf0 + buf1)[m0] = empty

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: array-splice, inst: buffer name: append-at2 seq: ⊦ # buf ≤ n ∧ n < # buf + # buf0 → (buf + buf0)[n] = buf0[n - # buf] subst: [n, buf0, buf] → [m0, buf1, buf0] flags: <all paths>

Additional information

Used simplifier rules: datasize-rec: ⊦ datasize(buf, n + 1) = (buf[n] = empty ⊃ datasize(buf, n);n + 1);% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ n < # buf → (buf + buf0)[n] = buf[n]; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)% ⊦ 1 = m + 1 ↔ m = 0; (s)% ⊦ m + 1 ≠ 0; (s) Used terms: m0, buf1, buf0