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Proof step 58 from proof infinite-append-lem (previous step)

¬ ii(m) + n < ii(n0) + # lf(n0) ∨ ii(m) + n < ii(n0), #((nf from ii(m)) to # lf(m)) = # lf(m) ∧ (∀ n. n < #((nf from ii(m)) to # lf(m)) → ((nf from ii(m)) to # lf(m))[n] = lf(m)[n]), ii(m + 1) = ii(m) + # lf(m), ii(0) = 0, n < # lf(m), m < n0, ∀ m0. m0 ≤ m → ii(m0) ≤ ii(m), ∀ m. m < n0 → ii(m + 1) = ii(m) + # lf(m) ∧ (nf from ii(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → ii(m0) ≤ ii(m)) ⊦ nf(ii(m) + n) = lf(m)[n], n0 = 0, n0 = m, m = n0 + 1, lf(m) = []

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 59:
ii(m + 1) = ii(m) + # lf(m), ii(0) = 0, n < # lf(m), m < n0, ¬ ii(m) + n < ii(n0) + # lf(n0) ∨ ii(m) + n < ii(n0), ∀ m0. m0 ≤ m → ii(m0) ≤ ii(m), ∀ m. m < n0 → ii(m + 1) = ii(m) + # lf(m) ∧ (nf from ii(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → ii(m0) ≤ ii(m)), ∀ n1. n1 < # lf(m) → nf(n1 + ii(m)) = lf(m)[n1] ⊦ nf(ii(m) + n) = lf(m)[n], n0 = 0, n0 = m, m = n0 + 1, lf(m) = []

Rule argument

Additional information

Used simplifier rules: get-from-inf: ⊦ ((nf from n))(m) = nf(m + n); from specification stream%get-infto: m < n ⊦ (nf to n)[m] = nf(m); from specification stream%ls-le-simp: ⊦ m < n → (n ≤ m ↔ false); from specification nat%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%size-inf-to: ⊦ #(nf to n) = n; from specification stream