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Proof step 69 from proof infinite-append-prefixnextli-weak (previous step)

#((σ from ii(n0)) to # nextli0(lf(n0))) = # ⌜nextli0(lf(n0))⌝ ∧ (∀ n. ⌜n⌝ < #((σ from ii(n0)) to # nextli0(lf(n0))) → ((σ from ii(n0)) to # nextli0(lf(n0)))[n] = ⌜nextli0(lf(n0))⌝ [n]), ii(0) = 0, lf(0) = [], # lf(n0) = ii(n0), n3 < # 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] ⊦ nextli0(lf(n0)) = [], ii(n0) + # nextli0(lf(n0)) = 0, nextli0(lf(n0))[n3] = σ[n3 + ii(n0)], # σ < ⌜ii(n0)⌝ + ⌜# nextli0(lf(n0))⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 70:
#((σ from ii(n0)) to # nextli0(lf(n0))) = ⌜# nextli0(lf(n0))⌝ , # lf(n0) = ii(n0), lf(0) = [], ii(0) = 0, n3 < # nextli0(lf(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))⌝ , ∀ n1. n1 < # nextli0(lf(n0)) → ((σ from ii(n0)) to # nextli0(lf(n0)))[n1] = nextli0(lf(n0))[n1] ⊦ nextli0(lf(n0))[n3] = σ[n3 + ii(n0)], ii(n0) + # nextli0(lf(n0)) = 0, nextli0(lf(n0)) = [], # σ < ⌜ii(n0)⌝ + ⌜# nextli0(lf(n0))⌝

Rule argument

Additional information

Used simplifier rules: less-fin: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; from specification nat-inf%sel-fin: ⊦ ⌜x⌝ [n] = x[n]; from specification stream%size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; from specification stream