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Proof step 51 from proof encode (previous step)

# buf5 = # buf4, # buf2 = # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), # buf0 = ENCODED_NAT_SIZE, # buf4 = (# buf2 + ENCODED_VOLID_SIZE + n2) + ENCODED_NAT_SIZE, # buf = # buf4, # buf3 = ENCODED_VOLID_SIZE + n2, subarray(buf5, 0, ENCODED_NAT_SIZE + # buf2) = buf0 + buf2, subarray(buf5, ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, n2) = subarray(buf3, ENCODED_VOLID_SIZE, n2), subarray(buf4, ENCODED_NAT_SIZE, # buf2 + ENCODED_VOLID_SIZE + n2) = subarray(buf, ENCODED_NAT_SIZE, # buf2 + ENCODED_VOLID_SIZE + n2), n2 < ENCODED_NAT_SIZE, # buf4 < 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, is-encoded(v, subarray(buf5, ENCODED_NAT_SIZE + # buf2, ENCODED_VOLID_SIZE)), is-encoded(#(vtbl1 ∪ vtbl), buf0), is-encoded-rec(vtbl, buf2), v ∈ vtbl1, v0 ∈ vtbl1, ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE ≤ # buf4, # vtbl1 * (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) ≤ ENCODED_VOLID_SIZE + n2, #(vtbl1 ∪ vtbl) * (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) ≤ # buf2 + ENCODED_VOLID_SIZE + n2, vtbl1 ⊥ vtbl, err0 = ESUCCESS → is-encoded(#(vtbl1 ∪ vtbl), subarray(buf4, 0, ENCODED_NAT_SIZE)) ∧ n1 = ENCODED_NAT_SIZE, err0 = ESUCCESS ∨ ⌊err0⌋ ⊦ v ∈ vtbl, v0 ∈ vtbl

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: store, inst: vtbl name: exin seq: ⊦ v ∈ vtbl ↔ (∃ n, vtbl0. vtbl = vtbl0[v, n] ∧ ¬ v ∈ vtbl0) subst: [vtbl, v] → [vtbl1, v0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ¬ v ∈ vtbl → # vtbl[v, n] = # vtbl + 1; (s)% ⊦ m * (n + 1) = m * n + m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ ¬ m + n < m; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ #(buf + buf0) = # buf + # buf0; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[encoding-nat-buffer]% ⊦ m ≤ m + n; (s)%# buf ≤ n0, n0 + m ≤ # buf + # buf0 ⊦ subarray(buf + buf0, n0, m) = subarray(buf0, n0 - # buf, m); (s) Used terms: vtbl1, v0