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

∀ x. nextli(x) = x + nextli0(x) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ¬ # σ < ⌜ii(n + 1)⌝ ∧ σ to ii(n + 1) = ⌜nextli((σ to ii(n)).l)⌝ ∧ (# σ = ⌜ii(n)⌝ ⊃ ii(n) = ii(n + 1);ii(n) < ii(n + 1))) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ ⌜nextli(σ.l)⌝ = σ ∧ (∀ m. (n ≤ m ⊃ ii(m) = ii(n);ii(m) < ii(m + 1))) ;∀ 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) = [] ∧ (∀ n. lf(n + 1) = nextli(lf(n)))
Proof step 6:
∃ lf. lf(0) = [] ∧ (∀ n. lf(n + 1) = nextli(lf(n))), ∀ x. nextli(x) = x + nextli0(x) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ¬ # σ < ⌜ii(n + 1)⌝ ∧ σ to ii(n + 1) = ⌜nextli((σ to ii(n)).l)⌝ ∧ (# σ = ⌜ii(n)⌝ ⊃ ii(n) = ii(n + 1);ii(n) < ii(n + 1))) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ ⌜nextli(σ.l)⌝ = σ ∧ (∀ m. (n ≤ m ⊃ ii(m) = ii(n);ii(m) < ii(m + 1))) ;∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) )

Rule argument

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

Additional information

Used terms: nextli