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Proof step 9 from proof nextl-nextj-lemma-butlast (previous step)

lfgen(0)(0) = a ' ∧ iigen(0)(0) = n0 ∧ (∀ n. n ≤ 0 → lfgen(0)(n + 1) = nextl(iigen(0)(n), lfgen(0)(n).last) ∧ iigen(0)(n + 1) = nextj(iigen(0)(n), lfgen(0)(n).last)), ∀ m. lfgen(m)(0) = a ' ∧ iigen(m)(0) = n0 ∧ (∀ n. n ≤ m → lfgen(m)(n + 1) = nextl(iigen(m)(n), lfgen(m)(n).last) ∧ iigen(m)(n + 1) = nextj(iigen(m)(n), lfgen(m)(n).last)), ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m) ⊦ lfgen(0)(0) = a ' ∧ iigen(0)(0) = n0 ∧ lfgen(n + 1)(n + 1) = nextl(iigen(n)(n), lfgen(n)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n)(n), lfgen(n)(n).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
lfgen(0)(0) = a ', iigen(0)(1) = nextj(iigen(0)(0), a), lfgen(0)(1) = nextl(iigen(0)(0), a), ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m), ∀ m. lfgen(m)(0) = a ' ∧ iigen(m)(0) = iigen(0)(0) ∧ (∀ n. n ≤ m → lfgen(m)(n + 1) = nextl(iigen(m)(n), lfgen(m)(n).last) ∧ iigen(m)(n + 1) = nextj(iigen(m)(n), lfgen(m)(n).last)) ⊦ lfgen(n + 1)(n + 1) = nextl(iigen(n)(n), lfgen(n)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n)(n), lfgen(n)(n).last)

Rule argument

Additional information

Used simplifier rules: l: ⊦ a '.last = a; from specification list-last%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%lez-01: ⊦ m ≤ 0 ↔ m = 0; from specification nat