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Proof step 3 from proof datasize-split-empty-h (previous step)

buf1 = mkempbuf(# buf1), m0 ≤ # buf1 ⊦ datasize(buf0 + buf1, # buf0 + m0) = datasize(buf0, # buf0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ m ≤ # buf0 + # buf1 → datasize(buf0 + buf1, m) = (m ≤ # buf0 ∨ datasize(buf1, m - # buf0) = 0 ⊃ datasize(buf0, min(m, # buf0));# buf0 + datasize(buf1, m - # buf0))
Proof step 5:
m ≤ # buf0 ⊦ datasize(buf0 + buf1, m) = datasize(buf0, m)
Proof step 6:
buf1 = mkempbuf(# buf1), m0 ≤ # buf1, # buf0 + m0 ≤ # buf0 + # buf1 ⊦ (# buf0 + m0 ≤ # buf0 ∨ datasize(buf1, # buf0 + m0 - # buf0) = 0 ⊃ datasize(buf0, min(# buf0 + m0, # buf0)) ;# buf0 + datasize(buf1, # buf0 + m0 - # buf0) ) = datasize(buf0, # buf0)

Rule argument

spec: <toplevel> name: datasize-append-h-new seq: ⊦ m ≤ # buf0 + # buf1 → datasize(buf0 + buf1, m) = (m ≤ # buf0 ∨ datasize(buf1, m - # buf0) = 0 ⊃ datasize(buf0, min(m, # buf0)) ;# buf0 + datasize(buf1, m - # buf0) ) subst: [m, buf1, buf0] → [# buf0 + m0, buf1, buf0] flags: <all paths>

Additional information

Used simplifier rules: datasize-append-lower-h: m ≤ # buf0 ⊦ datasize(buf0 + buf1, m) = datasize(buf0, m);% ⊦ m + m0 ≤ m ↔ m0 = 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m ≤ n → (n < m ↔ false); (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer] Used terms: # buf0 + m0, buf1, buf0