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Proof step 33 from proof infinite-append-lem (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 34:
⊦ nf from n = (λ m. nf(n + m))
Proof step 35:
ii(0) = 0, ii(m + 1) = ii(m) + # lf(m), (nf from ii(m)) to # lf(m) = lf(m), m < 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) ⊦ #((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]), m = n0 + 1, n0 = m, n0 = 0

Rule argument

spec: stream inst: name: from-nf seq: ⊦ nf from n = (λ m. nf(n + m)) subst: [n, nf] → [ii(m), λ n. (n < ii(n0) + # lf(n0) ∧ ¬ n < ii(n0) ⊃ lf(n0)[n - ii(n0)] ;nf(n) )] flags: rewrite

Additional information

Used terms: ii(m), λ n. (n < ii(n0) + # lf(n0) ∧ ¬ n < ii(n0) ⊃ lf(n0)[n - ii(n0)];nf(n))