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Proof step 32 from proof infinite-append-prefixnextli (previous step)

# σ = ⌜ii(n)⌝ , ii(0) = 0, ii(n0 + 1) = ii(n0), lf(0) = [], nextli0(lf(n0)) = [], finp(σ), ∀ m. n0 < m → ii(m) = ii(n0) ∧ lf(m) = lf(n0), ∀ 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))⌝ , ∀ n0. lf(n0) = (σ to ii(n0)).l, ∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n) ⊦ # σ = ⌜ii(n0)⌝ , # σ < ⌜ii(n0)⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 33:
# x = ii(n), ii(0) = 0, ii(n0 + 1) = ii(n0), lf(0) = [], nextli0(lf(n0)) = [], ii(n0) < ii(n), ∀ n1. ii(n1 + 1) = ii(n1) + # nextli0(lf(n1)) ∧ ¬ ii(n) < ii(n1 + 1) ∧ (x from ii(n1)) to # nextli0(lf(n1)) = nextli0(lf(n1)), ∀ n1. lf(n1) = x to ii(n1), ∀ m. n0 < m → ii(m) = ii(n0) ∧ lf(m) = lf(n0), ∀ x. nextli(x) = x + nextli0(x), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n) ⊦ ii(n) = ii(n0)

Rule argument

Additional information

Used simplifier rules: lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s)% ⊦ ⌜x⌝ from n = ⌜x from n⌝ ; (s)% ⊦ ⌜x⌝ .l = x; (s)% ⊦ ⌜x⌝ to n = ⌜x to n⌝ ; (s)% ⊦ finp(⌜x⌝ ); (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m ≠ n → (m < n ↔ ¬ n < m); (ss)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; (s)%stream-fin-forward: ⊦ finp(σ) → (∃ x. ⌜x⌝ = σ); (forward) from specification stream-basic