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Proof step 43 from proof infinite-append-lem (previous step)

ii(m) + n = n1 + ii(n0), ii(m1 + 1) = ii(m1) + # lf(m1), ii(m + 1) = ii(m) + # lf(m), ii(0) = 0, n0 = m1 + 1, (nf from ii(m1)) to # lf(m1) = lf(m1), (nf from ii(m)) to # lf(m) = lf(m), n1 < # lf(n0), n < # lf(m), m < n0, ∀ m0. m0 < n0 → ii(m0) ≤ ii(m1), ∀ 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)) ⊦ lf(n0)[n1] = lf(m)[n], lf(n0) = [], lf(m) = [], m = n0 + 1, n0 = m, n0 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 44:
ii(m) + n = n1 + ii(m1 + 1), ii(m1 + 1) = ii(m1) + # lf(m1), ii(m + 1) = ii(m) + # lf(m), ii(0) = 0, (nf from ii(m1)) to # lf(m1) = lf(m1), (nf from ii(m)) to # lf(m) = lf(m), n1 < # lf(m1 + 1), n < # lf(m), m ≤ m1, ∀ m0. ¬ m1 < m0 → ii(m0) ≤ ii(m1), ∀ m2. ¬ m1 < m2 → ii(m2 + 1) = ii(m2) + # lf(m2) ∧ (nf from ii(m2)) to # lf(m2) = lf(m2) ∧ (∀ m0. m0 ≤ m2 → ii(m0) ≤ ii(m2)), ∀ m0. m0 ≤ m → ii(m0) ≤ ii(m) ⊦ m1 + 1 = m, lf(m1 + 1)[n1] = lf(m)[n], lf(m1 + 1) = [], lf(m) = [], m = 2 + m1

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)