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Proof step 50 from proof infinite-append-prefixnextli (previous step)

ii(n0 + 1) = ii(n0), nextli0(lf(n0)) = [], ii(0) = 0, lf(0) = [], ii(m0) = ii(n0), lf(m0) = lf(n0), n0 < m0, ∀ n0. lf(n0) = (σ to ii(n0)).l, ii(m0 + 1) = ii(m0) + # nextli0(lf(m0)) ∧ ¬ # σ < ⌜ii(m0 + 1)⌝ ∧ (σ from ii(m0)) to # nextli0(lf(m0)) = ⌜nextli0(lf(m0))⌝ , ∀ 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)) ⊦ m0 = 0, # σ = ⌜ii(n0)⌝ , # σ < ⌜ii(n0)⌝ , ii(m0 + 1) = ii(n0) ∧ lf(m0 + 1) = lf(n0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 51:
lf(m0) = lf(n0), ii(m0) = ii(n0), lf(0) = [], ii(0) = 0, nextli0(lf(n0)) = [], ii(n0 + 1) = ii(n0), ii(m0 + 1) = ii(n0), n0 < m0, (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ 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 ⊦ # σ = ⌜ii(n0)⌝ , m0 = 0, lf(m0 + 1) = lf(n0), # σ < ⌜ii(n0)⌝

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%add-base: ⊦ n + 0 = n; from specification nat-basic2%empty: ⊦ ⌜[]⌝ = []; from specification stream%to-zero: ⊦ σ to 0 = []; from specification stream%size-01: ⊦ # [] = 0; from specification list-data