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Proof step 39 from proof infinite-append-lem (previous step)

ii(m) + n = n1 + ii(n0), ii(0) = 0, ii(m + 1) = ii(m) + # lf(m), (nf from ii(m)) to # lf(m) = lf(m), ii(n0) ≤ ii(m) + n, m < n0, n < # lf(m), ii(m) + n < ii(n0) + # lf(n0), ∀ 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)), ∀ m0. m0 ≤ m → ii(m0) ≤ ii(m) ⊦ ii(n0) + # lf(n0) = 0, lf(m) = [], m = n0 + 1, n0 = m, n0 = 0, lf(n0)[n1] = lf(m)[n], lf(n0) = []

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 40:
ii(m) + n = n1 + ii(n0), ii(0) = 0, ii(m + 1) = ii(m) + # lf(m), (nf from ii(m)) to # lf(m) = lf(m), m < n0, n < # lf(m), n1 < # lf(n0), ∀ 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)), ∀ m0. m0 ≤ m → ii(m0) ≤ ii(m) ⊦ ii(n0) + # lf(n0) = 0, lf(m) = [], m = n0 + 1, n0 = m, n0 = 0, lf(n0)[n1] = lf(m)[n], lf(n0) = []

Rule argument

Additional information

Used simplifier rules: l: ⊦ # x = 0 ↔ x = []; from specification list-data%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%lep: ⊦ m ≤ m + n; from specification nat%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%l: ⊦ m + n < m + n0 ↔ n < n0; from specification nat-basic2%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic1