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Proof step 48 from proof decode (previous step)

# buf1 = ENCODED_NAT_SIZE, # buf2 = ENCODED_VOLID_SIZE, # vtbl1 = m0 + 1, # buf0 = m0 * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), subarray(buf, ENCODED_NAT_SIZE + # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), # buf0 + ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) = buf2 + buf1 + buf0, is-encoded(v0, buf2), is-encoded(vtbl1[v0], buf1), is-encoded(m, subarray(buf, (ENCODED_NAT_SIZE + # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE)) + ENCODED_VOLID_SIZE, ENCODED_NAT_SIZE)), is-encoded(v, subarray(buf, ENCODED_NAT_SIZE + # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), ENCODED_VOLID_SIZE)), is-encoded(vtbl0, buf), is-encoded((m0 + 1) + # vtbl, subarray(buf, 0, ENCODED_NAT_SIZE)), is-encoded-rec(vtbl1 -- v0, buf0), is-encoded-rec(vtbl, subarray(buf, ENCODED_NAT_SIZE, # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE))), v0 ∈ vtbl1, 2 * ENCODED_NAT_SIZE + # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) + ENCODED_VOLID_SIZE ≤ # buf, ENCODED_NAT_SIZE + # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) + ENCODED_VOLID_SIZE ≤ # buf, 2 * ENCODED_NAT_SIZE + (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) * (m0 + # vtbl) + ENCODED_VOLID_SIZE ≤ # buf, vtbl ⊥ vtbl1, err0 = ESUCCESS → ENCODED_NAT_SIZE ≤ # buf ∧ n3 = ENCODED_NAT_SIZE, err0 = ESUCCESS ∨ ⌊err0⌋ ⊦ vtbl1 = ∅, v0 ∈ vtbl, # vtbl[v, m] = 1 + # vtbl ∧ #(vtbl1 -- v) = m0 ∧ ENCODED_NAT_SIZE + (#(vtbl1 -- v) + # vtbl[v, m]) * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) ≤ # buf ∧ is-encoded-rec(vtbl[v, m], subarray(buf, ENCODED_NAT_SIZE, # vtbl[v, m] * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE))) ∧ is-encoded-rec (vtbl1 -- v, subarray (buf, ENCODED_NAT_SIZE + # vtbl[v, m] * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), #(vtbl1 -- v) * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE)))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: array-splice, inst: encoding-nat-buffer name: subarray-elim seq: ⊦ n + m ≤ # buf → ( buf0 = subarray(buf, n, m) ↔ (∃ buf1, buf2. buf = buf1 + buf0 + buf2 ∧ # buf1 = n ∧ # buf0 = m)) subst: [buf0, m, n, buf] → [buf2 + buf1 + buf0, # buf0 + ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE, ENCODED_NAT_SIZE + # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), buf] flags: <all paths>

Additional information

Used simplifier rules: append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[encoding-nat-buffer]%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: buf2 + buf1 + buf0, # buf0 + ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE, ENCODED_NAT_SIZE + # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), buf