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Proof step 42 from proof infinite-append-prefixnextli (previous step)

lf(0) = [], ii(0) = 0, nextli0(lf(n0)) = [], ii(n0 + 1) = ii(n0), (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))⌝ , ∀ n0. lf(n0) = (σ to ii(n0)).l ⊦ ∀ m. n0 < m → ii(m) = ii(n0) ∧ lf(m) = lf(n0), # σ = ⌜ii(n0)⌝ , # σ < ⌜ii(n0)⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 43:
lf(0) = [], ii(0) = 0, nextli0(lf(n0)) = [], ii(n0 + 1) = ii(n0), n0 < m, (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))⌝ , ∀ n0. lf(n0) = (σ to ii(n0)).l ⊦ # σ = ⌜ii(n0)⌝ , m = 0, # σ < ⌜ii(n0)⌝ , ii(m) = ii(n0) ∧ lf(m) = lf(n0)

Rule argument

Additional information

Used simplifier rules: f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1