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Proof step 38 from proof infinite-append-prefixnextnli (previous step)

finp(σ), # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextnli0(lf(m), m) = [] ∧ ii(m) = ii(n)), ∀ 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 39:
ii(0) = 0, lf(0) = [], # x = ii(n), ∀ m. n ≤ m → nextnli0(lf(m), m) = [] ∧ ii(m) = ii(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), ∀ n0. lf(n0) = x to ii(n0), ∀ n0. ii(n0 + 1) = ii(n0) + # nextnli0(lf(n0), n0) ∧ ¬ ii(n) < ii(n0 + 1) ∧ (x from ii(n0)) to # nextnli0(lf(n0), n0) = nextnli0(lf(n0), n0) ⊦ ∃ n0. ii(n) = ii(n0) ∧ nextnli(x, n0) = x ∧ (∀ m. n0 ≤ m → ii(m) = ii(n0))

Rule argument

Additional information

Used simplifier rules: inj-02: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; from specification stream-basic%from-fin: ⊦ ⌜x⌝ from n = ⌜x from n⌝ ; from specification stream%less-fin: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; from specification nat-inf%sel-02: ⊦ ⌜x⌝ .l = x; from specification stream-basic%to-fin: ⊦ ⌜x⌝ to n = ⌜x to n⌝ ; from specification stream%test-02: ⊦ finp(⌜x⌝ ); from specification stream-basic%inj: ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; from specification nat-inf-basic%size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; from specification stream%stream-fin-forward: ⊦ finp(σ) → (∃ x. ⌜x⌝ = σ); (forward) from specification stream-basic