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Proof step 12 from proof datasize-empbuf-h (previous step)

(buf + mkempbuf(n))[m0] = empty, m0 < # buf + n, ∀ buf, n. m0 ≤ # buf + n → datasize(buf + mkempbuf(n), m0) = datasize(buf, min(# buf, m0)) ⊦ datasize(buf + mkempbuf(n), m0) = datasize(buf, (¬ m0 < # buf ⊃ # buf;m0 + 1))

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Proof step 13:
(buf + mkempbuf(n))[m0] = empty, m0 < # buf + n, ∀ buf, n. m0 ≤ # buf + n → datasize(buf + mkempbuf(n), m0) = datasize(buf, min(# buf, m0)) ⊦ datasize(buf + mkempbuf(n), m0) = datasize(buf, # buf), m0 < # buf
Proof step 20:
m ≤ # buf0 ⊦ datasize(buf0 + buf1, m) = datasize(buf0, m)
Proof step 21:
⊦ datasize(buf, n + 1) = (buf[n] = empty ⊃ datasize(buf, n);n + 1)

Rule argument

pos: right 1

Additional information

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