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Proof step 24 from proof is-encoded-rec-append (previous step)

# buf1 = (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) * m, # vtbl = m + 1, # buf0 = ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE, # buf3 = ENCODED_VOLID_SIZE, # buf2 = ENCODED_NAT_SIZE, is-encoded(n, subarray(buf0, ENCODED_VOLID_SIZE, ENCODED_NAT_SIZE)), is-encoded(v, subarray(buf0, 0, ENCODED_VOLID_SIZE)), is-encoded(vtbl[v0], buf2), is-encoded(v0, buf3), is-encoded-rec(vtbl -- v0, buf1), v0 ∈ vtbl, v0 ∈ vtbl[v, n], ∀ vtbl0, buf, buf0, v, n. # vtbl0 < m + 1 → # buf = # vtbl0 * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) ∧ # buf0 = ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE ∧ is-encoded(v, subarray(buf0, 0, ENCODED_VOLID_SIZE)) ∧ is-encoded(n, subarray(buf0, ENCODED_VOLID_SIZE, ENCODED_NAT_SIZE)) ∧ is-encoded-rec(vtbl0, buf) → is-encoded-rec(vtbl0[v, n], buf + buf0) ∨ v ∈ vtbl0 ⊦ vtbl = ∅, is-encoded-rec(vtbl[v, n] -- v0, buf1 + buf0), v ∈ vtbl

User applied rule apply induction

And obtained the following new goals

Proof step 25:
# buf2 = ENCODED_NAT_SIZE, # buf3 = ENCODED_VOLID_SIZE, # buf0 = ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE, # vtbl = m + 1, # buf1 = (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) * m, is-encoded(v0, buf3), is-encoded(vtbl[v0], buf2), is-encoded(v, subarray(buf0, 0, ENCODED_VOLID_SIZE)), is-encoded(n, subarray(buf0, ENCODED_VOLID_SIZE, ENCODED_NAT_SIZE)), is-encoded-rec(vtbl -- v0, buf1), v0 ∈ vtbl[v, n], v0 ∈ vtbl, is-encoded-rec((vtbl -- v0)[v, n], buf1 + buf0) ∨ v ∈ vtbl -- v0, ∀ vtbl0, buf, buf0, v, n. # vtbl0 < m + 1 → # buf = # vtbl0 * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) ∧ # buf0 = ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE ∧ is-encoded(v, subarray(buf0, 0, ENCODED_VOLID_SIZE)) ∧ is-encoded(n, subarray(buf0, ENCODED_VOLID_SIZE, ENCODED_NAT_SIZE)) ∧ is-encoded-rec(vtbl0, buf) → is-encoded-rec(vtbl0[v, n], buf + buf0) ∨ v ∈ vtbl0 ⊦ vtbl = ∅, is-encoded-rec(vtbl[v, n] -- v0, buf1 + buf0), v ∈ vtbl

Rule argument

subst: [vtbl0, buf, buf0, v, n] → [vtbl -- v0, buf1, buf0, v, n]

Additional information

Used simplifier rules: is-encoded-rec-unique: ⊦ is-encoded-rec(vtbl0, buf) → (is-encoded-rec(vtbl1, buf) ↔ vtbl0 = vtbl1);% ⊦ v ∈ vtbl → #(vtbl -- v) = (# vtbl)-1; (s)% ⊦ n -1 = n - 1; (s)% ⊦ is-encoded(v, buf) → (is-encoded(v0, buf) ↔ v = v0); (s)% ⊦ is-encoded(n, buf) → (is-encoded(n0, buf) ↔ n = n0); (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ ¬ m < n → ¬ m + m0 < n; (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) Used terms: vtbl -- v0, buf1, buf0, v, n