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Proof step 4 from proof infinite-append-prefixnextnli (previous step)

∀ x, n. nextnli(x, n) = x + nextnli0(x, n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n) ≤ ii(n + 1) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ σ to ii(n + 1) = ⌜nextnli((σ to ii(n)).l, n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ ⌜nextnli(σ.l, n)⌝ = σ ∧ (∀ m. n ≤ m → ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

User applied rule insert lemma

And obtained the following new goals

Proof step 5:
⊦ ∃ lf. lf(0) = x ∧ (∀ n. lf(n + 1) = nextnli(lf(n), n))
Proof step 6:
∃ lf. lf(0) = [] ∧ (∀ n. lf(n + 1) = nextnli(lf(n), n)), ∀ x, n. nextnli(x, n) = x + nextnli0(x, n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n) ≤ ii(n + 1) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ σ to ii(n + 1) = ⌜nextnli((σ to ii(n)).l, n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ ⌜nextnli(σ.l, n)⌝ = σ ∧ (∀ m. n ≤ m → ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

spec: inst: name: nextnli-lemma seq: ⊦ ∃ lf. lf(0) = x ∧ (∀ n. lf(n + 1) = nextnli(lf(n), n)) subst: [x, nextnli] → [[], nextnli] flags: current, []

Additional information

Used terms: [], nextnli