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Proof step 73 from proof infinite-append-prefixnextli-weak (previous step)

# lf(n0) = ii(n0), #((σ from ii(n0)) to # nextli0(lf(n0))) = ⌜# nextli0(lf(n0))⌝ , ii(0) = 0, lf(0) = [], n3 < # nextli0(lf(n0)), ∀ n1. n1 < # nextli0(lf(n0)) → ((σ from ii(n0)) to # nextli0(lf(n0)))[n1] = nextli0(lf(n0))[n1], ∀ 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), ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2], ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ⊦ nextli0(lf(n0))[n3] = σ[n3 + ii(n0)], nextli0(lf(n0)) = [], # σ < ⌜ii(n0)⌝ + ⌜# nextli0(lf(n0))⌝ , finp(σ)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 74:
# lf(n0) = ii(n0), ii(0) = 0, lf(0) = [], n3 < # nextli0(lf(n0)), ∀ 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. n < # nextli0(lf(n0)) → nf(n + ii(n0)) = nextli0(lf(n0))[n], ∀ n. n < ii(n0) → lf(n0)[n] = nf(n), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ (nf from ii(n)) to # nextli0(lf(n)) = nextli0(lf(n)) ⊦ nextli0(lf(n0))[n3] = nf(n3 + ii(n0)), nextli0(lf(n0)) = []

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s)% ⊦ ⌜nf⌝ to n = ⌜nf to n⌝ ; (s)% ⊦ ((nf from n))(m) = nf(m + n); (s)% ⊦ ¬ finp(⌜nf⌝ ); (s)% ⊦ ⌜nf⌝ [n] = nf(n); (s)%m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n)[m] = σ[m]; (s)% ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; (s)%¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; (s)% ⊦ # ⌜nf⌝ = ∞; (s)% ⊦ ⌜nf⌝ from n = ⌜nf from n⌝ ; (s)%stream-inf-forward: ⊦ ¬ finp(σ) → (∃ nf. ⌜nf⌝ = σ); (forward) from specification stream-basic% ⊦ ¬ ∞ < n∞; (s)% ⊦ ¬ finp(σ) → # σ = ∞; (s)