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Proof step 75 from proof infinite-append-prefixnextli-weak (previous step)

n1 < # lf(n0), (σ from ii(n0)) to # nextli0(lf(n0)) = ⌜nextli0(lf(n0))⌝ , ii(0) = 0, lf(0) = [], # lf(n0) = ii(n0), n1 < ii(n0) + # nextli0(lf(n0)), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ x. nextli(x) = x + nextli0(x), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2] ⊦ lf(n0)[n1] = σ[n1], ii(n0) + # nextli0(lf(n0)) = 0, # σ < ⌜ii(n0)⌝ + ⌜# nextli0(lf(n0))⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 76:
# lf(n0) = ii(n0), lf(0) = [], ii(0) = 0, (σ from ii(n0)) to # nextli0(lf(n0)) = ⌜nextli0(lf(n0))⌝ , n1 < ii(n0) + # nextli0(lf(n0)), n1 < ii(n0), ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2], (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ x. nextli(x) = x + nextli0(x), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ ⊦ ii(n0) = 0, ii(n0) + # nextli0(lf(n0)) = 0, lf(n0)[n1] = σ[n1], # σ < ⌜ii(n0)⌝ + ⌜# nextli0(lf(n0))⌝

Rule argument

Additional information

Used simplifier rules: lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic1%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1