[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

Rule argument

Additional information

Used simplifier rules: ⊦ ((nf from n))(m) = nf(m + n); (s)%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)