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Proof step 84 from proof infinite-append-prefixnextnli (previous step)

#((σ from ii(n0)) to # nextnli0(lf(n0), n0)) = ⌜# nextnli0(lf(n0), n0)⌝ , # lf(n0) = ii(n0), ii(0) = 0, lf(0) = [], n1 = n3 + ii(n0), n1 < ii(n0) + # nextnli0(lf(n0), n0), ii(n0) ≤ n1, (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextnli0(lf(m), m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2], ∀ n2. n2 < # nextnli0(lf(n0), n0) → ((σ from ii(n0)) to # nextnli0(lf(n0), n0))[n2] = nextnli0(lf(n0), n0)[n2] ⊦ nextnli0(lf(n0), n0)[n3] = σ[n1], nextnli0(lf(n0), n0) = [], # σ < ⌜ii(n0)⌝ + ⌜# nextnli0(lf(n0), n0)⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 85:
#((σ from ii(n0)) to # nextnli0(lf(n0), n0)) = ⌜# nextnli0(lf(n0), n0)⌝ , # lf(n0) = ii(n0), ii(0) = 0, lf(0) = [], n3 < # nextnli0(lf(n0), n0), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextnli0(lf(m), m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2], ∀ n2. n2 < # nextnli0(lf(n0), n0) → ((σ from ii(n0)) to # nextnli0(lf(n0), n0))[n2] = nextnli0(lf(n0), n0)[n2] ⊦ nextnli0(lf(n0), n0)[n3] = σ[n3 + ii(n0)], nextnli0(lf(n0), n0) = [], # σ < ⌜ii(n0)⌝ + ⌜# nextnli0(lf(n0), n0)⌝

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m + n < m + n0 ↔ n < n0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ m + n; (s)