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Proof step 12 from proof datasize-append-h-new (previous step)

n +1 ≤ # buf1, ∀ buf1, buf0. n ≤ # buf1 → # buf1 = 0 ∨ datasize(buf1, n) = 0 ∨ datasize(buf0 + buf1, n + # buf0) = # buf0 + datasize(buf1, n) ∨ n = 0 ⊦ # buf1 = 0, n +1 = 0, datasize(buf0 + buf1, n +1 + # buf0) = # buf0 + datasize(buf1, n +1), datasize(buf1, n +1) = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
⊦ datasize(buf, n + 1) = (buf[n] = empty ⊃ datasize(buf, n);n + 1)
Proof step 14:
n < # buf1, ∀ buf1, buf0. n ≤ # buf1 → # buf1 = 0 ∨ datasize(buf1, n) = 0 ∨ datasize(buf0 + buf1, n + # buf0) = # buf0 + datasize(buf1, n) ∨ n = 0 ⊦ (buf1[n] = empty ⊃ datasize(buf0 + buf1, n + # buf0);(n + # buf0) + 1) = # buf0 + (buf1[n] = empty ⊃ datasize(buf1, n);n + 1), (buf1[n] = empty ⊃ datasize(buf1, n);n + 1) = 0

Rule argument

Additional information

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