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Proof step 83 from proof infinite-append-prefixnextnli (previous step)

# lf(n0) = ii(n0), lf(0) = [], ii(0) = 0, #((σ from ii(n0)) to # nextnli0(lf(n0), n0)) = ⌜# nextnli0(lf(n0), n0)⌝ , ii(n0) ≤ n1, n1 < ii(n0) + # nextnli0(lf(n0), n0), ∀ n2. n2 < # nextnli0(lf(n0), n0) → ((σ from ii(n0)) to # nextnli0(lf(n0), n0))[n2] = nextnli0(lf(n0), n0)[n2], ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2], (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) ⊦ nextnli0(lf(n0), n0)[n1 - ii(n0)] = σ[n1], ii(n0) + # nextnli0(lf(n0), n0) = 0, nextnli0(lf(n0), n0) = [], # σ < ⌜ii(n0)⌝ + ⌜# nextnli0(lf(n0), n0)⌝

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 84:
n1 = n3 + ii(n0), # lf(n0) = ii(n0), lf(0) = [], ii(0) = 0, #((σ from ii(n0)) to # nextnli0(lf(n0), n0)) = ⌜# nextnli0(lf(n0), n0)⌝ , ii(n0) ≤ n1, n1 < ii(n0) + # nextnli0(lf(n0), n0), ∀ n2. n2 < # nextnli0(lf(n0), n0) → ((σ from ii(n0)) to # nextnli0(lf(n0), n0))[n2] = nextnli0(lf(n0), n0)[n2], ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2], (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) ⊦ nextnli0(lf(n0), n0)[n3] = σ[n1], ii(n0) + # nextnli0(lf(n0), n0) = 0, nextnli0(lf(n0), n0) = [], # σ < ⌜ii(n0)⌝ + ⌜# nextnli0(lf(n0), n0)⌝

Rule argument

spec: nat inst: name: elim-sub seq: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n subst: [m, n, n0] → [n1, ii(n0), n3] flags:

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n1, ii(n0), n3