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Proof step 5 from proof lebs-write-above (previous step)

n + m0 < # avol, n + m0 < n0, ∀ n, n0, avol, aleb. n + m0 ≤ n0 ∧ n + m0 ≤ # avol → lebs(avol[n0, aleb], n, m0) = lebs(avol, n, m0) ⊦ lebs(avol[n0, aleb], n, m0 + 1) = lebs(avol, n, m0) + (avol[n + m0].mapped? ⊃ avol[n + m0].data + mkempbuf(LEB_SIZE - # avol[n + m0].data);mkempbuf(LEB_SIZE))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: lebs-rec seq: ⊦ n + m < # avol → lebs(avol, n, m + 1) = lebs(avol, n, m) + (avol[n + m].mapped? ⊃ avol[n + m].data + mkempbuf(LEB_SIZE - # avol[n + m].data) ;mkempbuf(LEB_SIZE) ) subst: [m, n, avol] → [m0, n, avol[n0, aleb]] flags: <all paths>

Additional information

Used simplifier rules: lebs-write-below: n0 < n, n + m ≤ # avol ⊦ lebs(avol[n0, aleb], n, m) = lebs(avol, n, m);%lebs-rec: ⊦ n + m < # avol → lebs(avol, n, m + 1) = lebs(avol, n, m) + (avol[n + m].mapped? ⊃ avol[n + m].data + mkempbuf(LEB_SIZE - # avol[n + m].data) ;mkempbuf(LEB_SIZE) );% ⊦ m + 1 ≤ n ↔ m < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ # avol[n, aleb] = # avol; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ buf + buf0 = buf + buf1 ↔ buf0 = buf1; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer] Used terms: m0, n, avol[n0, aleb]