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Proof step 69 from proof encode (previous step)

# buf = 0, # buf1 = 0, # vtbl1 = m + 1, # buf7 = ENCODED_NAT_SIZE, # buf4 = n0 + 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, # buf6 = # buf4, # buf5 = # buf4, # buf2 = # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), # buf0 = ENCODED_NAT_SIZE, # buf8 = ENCODED_VOLID_SIZE, subarray(buf6, 0, (ENCODED_NAT_SIZE + # buf2) + ENCODED_VOLID_SIZE) = subarray(buf5, 0, (ENCODED_NAT_SIZE + # buf2) + ENCODED_VOLID_SIZE), subarray(buf6, 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, n0) = subarray(buf5, 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, n0), subarray(buf5, 0, ENCODED_NAT_SIZE + # buf2) = buf0 + buf2, is-encoded(v, subarray(buf5, ENCODED_NAT_SIZE + # buf2, ENCODED_VOLID_SIZE)), is-encoded(#(vtbl1 ∪ vtbl), buf0), is-encoded(vtbl1[v], subarray(buf6, (ENCODED_NAT_SIZE + # buf2) + ENCODED_VOLID_SIZE, ENCODED_NAT_SIZE)), is-encoded-rec(vtbl, buf2), v ∈ vtbl1, 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE ≤ # buf4, ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE ≤ # buf4, (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) * m ≤ n0, #(vtbl1 ∪ vtbl) * (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) ≤ # buf2 + ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE + n0, vtbl1 ⊥ vtbl, err0 = ESUCCESS → is-encoded(#(vtbl1 ∪ vtbl), subarray(buf4, 0, ENCODED_NAT_SIZE)) ∧ n1 = ENCODED_NAT_SIZE, err0 = ESUCCESS ∨ ⌊err0⌋ ⊦ vtbl1 = ∅, v ∈ vtbl, ∃ buf1, buf7, buf8, vtbl0. buf6 = buf1 + buf7 + buf8 ∧ vtbl1 ∪ vtbl = (vtbl1 -- v) ∪ vtbl0 ∧ vtbl1 -- v ⊥ vtbl0 ∧ # buf1 + # buf7 = 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE ∧ m * (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) ≤ # buf8 ∧ is-encoded(#((vtbl1 -- v) ∪ vtbl0), buf1) ∧ is-encoded-rec(vtbl0, buf7)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 70:
⊦ n + m ≤ # buf → (buf0 = subarray(buf, n, m) ↔ (∃ buf1, buf2. buf = buf1 + buf0 + buf2 ∧ # buf1 = n ∧ # buf0 = m))
Proof step 71:
# buf8 = ENCODED_VOLID_SIZE, # buf0 = ENCODED_NAT_SIZE, # buf2 = # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), # buf5 = # buf4, # buf6 = # buf4, # buf4 = n0 + 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, # buf7 = ENCODED_NAT_SIZE, # vtbl1 = m + 1, # buf1 = 0, # buf = 0, subarray(buf5, 0, ENCODED_NAT_SIZE + # buf2) = buf0 + buf2, subarray(buf6, 0, (ENCODED_NAT_SIZE + # buf2) + ENCODED_VOLID_SIZE) = subarray(buf5, 0, (ENCODED_NAT_SIZE + # buf2) + ENCODED_VOLID_SIZE), is-encoded(vtbl1[v], subarray(buf6, (ENCODED_NAT_SIZE + # buf2) + ENCODED_VOLID_SIZE, ENCODED_NAT_SIZE)), is-encoded(#(vtbl1 ∪ vtbl), buf0), is-encoded(v, subarray(buf5, ENCODED_NAT_SIZE + # buf2, ENCODED_VOLID_SIZE)), is-encoded-rec(vtbl, buf2), v ∈ vtbl1, #(vtbl1 ∪ vtbl) * (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) ≤ # buf2 + ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE + n0, (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) * m ≤ n0, ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE ≤ # buf4, 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE ≤ # buf4, (2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE) + n0 ≤ # buf5, vtbl1 ⊥ vtbl, err0 = ESUCCESS → is-encoded(#(vtbl1 ∪ vtbl), subarray(buf4, 0, ENCODED_NAT_SIZE)) ∧ n1 = ENCODED_NAT_SIZE, ∃ buf1, buf3. buf5 = buf1 + subarray(buf6, 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, n0) + buf3 ∧ # buf1 = 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE ∧ # subarray(buf6, 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, n0) = n0, err0 = ESUCCESS ∨ ⌊err0⌋ ⊦ vtbl1 = ∅, v ∈ vtbl, ∃ buf1, buf7, buf8, vtbl0. buf6 = buf1 + buf7 + buf8 ∧ vtbl1 ∪ vtbl = (vtbl1 -- v) ∪ vtbl0 ∧ vtbl1 -- v ⊥ vtbl0 ∧ # buf1 + # buf7 = 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE ∧ m * (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) ≤ # buf8 ∧ is-encoded(#((vtbl1 -- v) ∪ vtbl0), buf1) ∧ is-encoded-rec(vtbl0, buf7)

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: [m, n, buf, buf0] → [n0, 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, buf5, subarray (buf6, 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, n0)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m ≤ n ↔ ¬ n < m; (ss)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%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% ⊦ m < n → n ≠ 0; (s)% ⊦ m < n → ¬ n < m; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s) Used terms: n0, 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, buf5, subarray(buf6, 2 * ENCODED_NAT_SIZE + # buf2 + ENCODED_VOLID_SIZE, n0)