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Proof step 51 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, (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) * (m0 + # vtbl) < # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) + # buf0, # buf < 2 * ENCODED_NAT_SIZE + # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) + # buf0 + ENCODED_VOLID_SIZE, 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⌋ ⊦ # buf0 = 0, ENCODED_NAT_SIZE + # buf0 = 0, 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: nat-mult name: dist seq: ⊦ m0 * (m + n) = m0 * m + m0 * n subst: [n, m, m0] → [# vtbl, m0, ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE] flags: <all paths>

Additional information

Used simplifier rules: mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ m + n < m + n0 ↔ n < n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n → n ≠ 0; (s)%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%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ ¬ n < n; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)%is-encoded-size: ⊦ is-encoded(n, buf) → # buf = ENCODED_NAT_SIZE; (forward) from specification encoding-fixed-size[encoding-nat]%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ m < n → ¬ n < m; (s) Used terms: # vtbl, m0, ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE