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Proof step 18 from proof is-encoded-rec-unique (previous step)

# buf2 = ENCODED_VOLID_SIZE, # buf1 = ENCODED_NAT_SIZE, # buf0 = # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE + # buf0 = # vtbl1 * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), # vtbl + 1 = # vtbl1, is-encoded(vtbl[v, n][v0], buf1), is-encoded(v0, buf2), is-encoded-rec(vtbl1, buf2 + buf1 + buf0), is-encoded-rec(vtbl[v, n] -- v0, buf0), v0 ∈ vtbl[v, n], ∀ vtbl2, buf, vtbl1. # vtbl2 < # vtbl[v, n] → # buf = # vtbl1 * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) ∧ # buf = # vtbl2 * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) ∧ is-encoded-rec(vtbl2, buf) ∧ is-encoded-rec(vtbl1, buf) → vtbl2 = vtbl1 ⊦ ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE = 0, vtbl[v, n] = vtbl1, v ∈ vtbl

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 19:
vtbl ≠ ∅ ⊦ is-encoded-rec(vtbl, buf) ↔ (∃ v, buf0, buf1, buf2. v ∈ vtbl ∧ buf = buf0 + buf1 + buf2 ∧ is-encoded(v, buf0) ∧ is-encoded(vtbl[v], buf1) ∧ is-encoded-rec(vtbl -- v, buf2))
Proof step 20:
# buf0 = # vtbl * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), # buf1 = ENCODED_NAT_SIZE, # buf2 = ENCODED_VOLID_SIZE, # vtbl + 1 = # vtbl1, ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE + # buf0 = # vtbl1 * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE), is-encoded(v0, buf2), is-encoded(vtbl[v, n][v0], buf1), is-encoded-rec(vtbl[v, n] -- v0, buf0), v0 ∈ vtbl[v, n], ∃ v, buf5, buf4, buf3. v ∈ vtbl1 ∧ buf2 + buf1 + buf0 = buf5 + buf4 + buf3 ∧ is-encoded(v, buf5) ∧ is-encoded(vtbl1[v], buf4) ∧ is-encoded-rec(vtbl1 -- v, buf3), ∀ vtbl2, buf, vtbl1. # vtbl2 < # vtbl[v, n] → # buf = # vtbl1 * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) ∧ # buf = # vtbl2 * (ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE) ∧ is-encoded-rec(vtbl2, buf) ∧ is-encoded-rec(vtbl1, buf) → vtbl2 = vtbl1 ⊦ ENCODED_VOLID_SIZE + ENCODED_NAT_SIZE = 0, vtbl[v, n] = vtbl1, vtbl1 = ∅, v ∈ vtbl

Rule argument

spec: <toplevel> name: is-encoded-rec seq: vtbl ≠ ∅ ⊦ is-encoded-rec(vtbl, buf) ↔ (∃ v, buf0, buf1, buf2. v ∈ vtbl ∧ buf = buf0 + buf1 + buf2 ∧ is-encoded(v, buf0) ∧ is-encoded(vtbl[v], buf1) ∧ is-encoded-rec(vtbl -- v, buf2)) subst: [buf, vtbl] → [buf2 + buf1 + buf0, vtbl1] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # ∅ = 0; (s)% ⊦ 0 * n = 0; (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 = 0 ↔ m = 0 ∧ n = 0; (ws)% ⊦ ENCODED_VOLID_SIZE ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: buf2 + buf1 + buf0, vtbl1