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Proof step 51 from proof infinite-append-prefixnextnli (previous step)

¬ finp(σ), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. lf(n0) = (σ to ii(n0)).l, ii(0) = 0, lf(0) = [], ∀ n. ii(n + 1) = ii(n) + # nextnli0(lf(n), n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextnli0(lf(n), n) = ⌜nextnli0(lf(n), n)⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextnli(lf(n), n), ∀ x, n. nextnli(x, n) = x + nextnli0(x, n) ⊦ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ ⌜nextnli(σ.l, n)⌝ = σ ∧ (∀ m. n ≤ m → ii(m) = ii(n));∃ m. ii(m) ≤ n1 ∧ n1 < ii(m + 1))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 52:
ii(0) = 0, lf(0) = [], ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextnli(lf(n), n), ∀ x, n. nextnli(x, n) = x + nextnli0(x, n), ∀ n. lf(n) = nf to ii(n), ∀ n. ii(n + 1) = ii(n) + # nextnli0(lf(n), n) ∧ (nf from ii(n)) to # nextnli0(lf(n), n) = nextnli0(lf(n), 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