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Proof step 188 from proof infinite-append-prefixnextli (previous step)

¬ finp(σ), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), #((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], ∀ 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))⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: inj-02: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; from specification stream-basic%to-inf: ⊦ ⌜nf⌝ to n = ⌜nf to n⌝ ; from specification stream%get-from-inf: ⊦ ((nf from n))(m) = nf(m + n); from specification stream%test-01: ⊦ ¬ finp(⌜nf⌝ ); from specification stream-basic%sel-inf: ⊦ ⌜nf⌝ [n] = nf(n); from specification stream%get-to: m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n)[m] = σ[m]; from specification stream%inj: ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; from specification nat-inf-basic%size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; from specification stream%size-inf: ⊦ # ⌜nf⌝ = ∞; from specification stream%from-inf: ⊦ ⌜nf⌝ from n = ⌜nf from n⌝ ; from specification stream%stream-inf-forward: ⊦ ¬ finp(σ) → (∃ nf. ⌜nf⌝ = σ); (forward) from specification stream-basic%less-inf1: ⊦ ¬ ∞ < n∞; from specification nat-inf%notfin-sizeinf: ⊦ ¬ finp(σ) → # σ = ∞; from specification stream