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Proof step 38 from proof is-encoded-split-one (previous step)

# ns = m + 1, # buf1 = ENCODED_NAT_SIZE, # buf2 = ENCODED_NAT_SIZE, # buf = m * ENCODED_NAT_SIZE, m + 1 < #((ns -- n0) ++ n), is-encoded(n, buf1), is-encoded(ns -- n0, buf), is-encoded(n0, buf2), n0 ∈ ns, ∀ ns0, buf0, ns2, buf1, n0. # ns0 < #(ns ++ n) → # buf0 = # ns2 * ENCODED_NAT_SIZE ∧ # buf1 = ENCODED_NAT_SIZE ∧ is-encoded(n0, buf1) ∧ is-encoded(ns2, buf0) → (is-encoded(ns0, buf0 + buf1) ↔ ns0 = ns2 ++ n0) ∨ n0 ∈ ns2 ⊦ ns = ∅, n ∈ ns, ∃ n0, buf4, buf3. (n0 = n ∨ n0 ∈ ns) ∧ (buf2 + buf) + buf1 = buf4 + buf3 ∧ is-encoded(n0, buf4) ∧ is-encoded((ns ++ n) -- n0, buf3)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: set-basic, inst: nat-set name: Size-insert seq: ⊦ ¬ n ∈ ns → #(ns ++ n) = # ns + 1 subst: [n, ns] → [n, ns -- n0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n ∈ ns → #(ns -- n) = # ns - 1; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ ¬ m < n → ¬ m + m0 < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%n0 ≤ n ⊦ m < n - n0 ↔ m + n0 < n; (s)% ⊦ n ∈ ns -- n0 ↔ n ≠ n0 ∧ n ∈ ns; (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% ⊦ m + 1 ≠ 0; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s) Used terms: n, ns -- n0