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Proof step 35 from proof infinite-append-lem (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 36:
ii(m + 1) = ii(m) + # lf(m), ii(0) = 0, (nf from ii(m)) to # lf(m) = lf(m), 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)) ⊦ (ii(m) + n < ii(n0) + # lf(n0) ∧ ¬ ii(m) + n < ii(n0) ⊃ lf(n0)[ii(m) + n - ii(n0)];nf(ii(m) + n)) = lf(m)[n], lf(m) = [], m = n0 + 1, n0 = m

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%m < n ⊦ (nf to n)[m] = nf(m); (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ #(nf to n) = n; (s)