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Proof step 33 from proof infinite-append-prefixnextnli (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 34:
#(x to ii(n0 + 1)) = # lf(n0 + 1), lf(n0 + 1) = lf(n0) + nextnli0(lf(n0), n0), lf(0) = [], ii(0) = 0, lf(n0) = x to ii(n0), # x = ii(n), n1 < # lf(n0 + 1), ∀ n2. ii(n2 + 1) = ii(n2) + # nextnli0(lf(n2), n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ (x from ii(n2)) to # nextnli0(lf(n2), n2) = nextnli0(lf(n2), n2), ∀ n2. lf(n2) = x to ii(n2), ∀ n2. n2 < # lf(n0 + 1) → (x to ii(n0 + 1))[n2] = lf(n0 + 1)[n2], ∀ x, n. nextnli(x, n) = x + nextnli0(x, n), ∀ n. lf(n + 1) = nextnli(lf(n), n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → nextnli0(lf(m), m) = [] ∧ ii(m) = ii(n) ⊦ (x to ii(n0 + 1))[n1] = lf(n0 + 1)[n1], lf(n0 + 1) = []

Rule argument

Additional information

Used simplifier rules: l: ⊦ # x = 0 ↔ x = []; from specification list-data%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic1%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%sel-fin: ⊦ ⌜x⌝ [n] = x[n]; from specification stream%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%test-02: ⊦ finp(⌜x⌝ ); from specification stream-basic%sel-02: ⊦ ⌜x⌝ .l = x; from specification stream-basic%to-fin: ⊦ ⌜x⌝ to n = ⌜x to n⌝ ; from specification stream%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