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Proof step 78 from proof decode (previous step)

# buf0 = m0 * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), # vtbl1 = m0 + 1, # buf2 = ENCODED_VOLID_SIZE, # buf1 = ENCODED_NAT_SIZE, ENCODED_NAT_SIZE + (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) * # vtbl = # buf4, is-encoded((m0 + 1) + # vtbl, subarray(buf4, 0, ENCODED_NAT_SIZE)), is-encoded(vtbl0, (buf4 + buf2 + buf1) + buf0 + buf3), is-encoded(vtbl1[v], buf1), is-encoded(v, buf2), is-encoded-rec(vtbl, subarray(buf4, ENCODED_NAT_SIZE, # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE))), is-encoded-rec(vtbl1 -- v, buf0), is-encoded-rec(vtbl[v, vtbl1[v]], subarray(buf4, ENCODED_NAT_SIZE, (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) * # vtbl) + buf2 + buf1), v ∈ vtbl1, ENCODED_NAT_SIZE + (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) * (m0 + # vtbl) ≤ # buf4 + # buf0 + # buf3, # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) ≤ # buf4 + # buf0 + # buf3, vtbl ⊥ vtbl1, err0 = ESUCCESS → n3 = ENCODED_NAT_SIZE, err0 = ESUCCESS ∨ ⌊err0⌋ ⊦ vtbl1 = ∅, is-encoded-rec (vtbl[v, vtbl1[v]], subarray(buf4 + buf2 + buf1, ENCODED_NAT_SIZE, (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) * # vtbl + ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE)), v ∈ vtbl

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: array-splice, inst: encoding-nat-buffer name: subarray-append-shift-left seq: ⊦ n + m = # buf + # buf0 ∧ n ≤ # buf → subarray(buf + buf0, n, m) = subarray(buf, n, # buf - n) + buf0 subst: [m, n, buf0, buf] → [ (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) * # vtbl + ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE, ENCODED_NAT_SIZE, buf2 + buf1, buf4] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + m = 2 * m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; 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 + n = m + n0 ↔ n = n0; (s)%m2 ≠ 0 ⊦ (* m2) * m + n = m + n0 ↔ (m2 - 1) * m + n = n0; (s)% ⊦ m0 * 1 = m0; (s)%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 ≤ n ↔ m < n); (s)% ⊦ m ≠ n → (m < n ↔ ¬ n < m); (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ ENCODED_NAT_SIZE ≠ 0; (s)% ⊦ ¬ m + n < m; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m < n → m ≤ n; (s)%is-encoded-size: ⊦ is-encoded(n, buf) → # buf = ENCODED_NAT_SIZE; (forward) from specification encoding-fixed-size[encoding-nat]% ⊦ m < n → n ≠ 0; (s)%# buf ≤ n0, n0 + m ≤ # buf + # buf0 ⊦ subarray(buf + buf0, n0, m) = subarray(buf0, n0 - # buf, m); (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)% ⊦ m + m0 ≤ m ↔ m0 = 0; (s)% ⊦ ENCODED_VOLID_SIZE ≠ 0; (s)% ⊦ ⌊err⌋ → err ≠ ESUCCESS; (s) Used terms: (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) * # vtbl + ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE, ENCODED_NAT_SIZE, buf2 + buf1, buf4