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Proof step 151 from proof infinite-append-prefixnextli (previous step)

¬ finp(σ), ∀ n0. lf(n0) = (σ to ii(n0)).l, ii(0) = 0, lf(0) = [], ∀ 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)) ⊦ ∃ m. ii(m) ≤ n1 ∧ n1 < ii(m + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 152:
ii(0) = 0, lf(0) = [], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ x. nextli(x) = x + nextli0(x), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. lf(n) = nf to ii(n), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ (nf from ii(n)) to # nextli0(lf(n)) = nextli0(lf(n)) ⊦ ∃ m. ii(m) ≤ n1 ∧ n1 < ii(m + 1)

Rule argument

Additional information

Used simplifier rules: inj-02: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; from specification stream-basic%from-inf: ⊦ ⌜nf⌝ from n = ⌜nf from n⌝ ; from specification stream%sel-02: ⊦ ⌜x⌝ .l = x; from specification stream-basic%to-inf: ⊦ ⌜nf⌝ to n = ⌜nf to n⌝ ; from specification stream%test-01: ⊦ ¬ finp(⌜nf⌝ ); from specification stream-basic%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