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Proof step 5 from proof heap-order-add-smaller (previous step)

m0 < # kar, kar[n, ki][m0 + 1].key < kar[m0 / 2].key, m0 < n, n < # kar, m0 + 1 < # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ ki.key < kar[(n - 1) / 2].key

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 6:
n = m1 + 1, m0 + 1 < # kar, n < # kar, m0 < n, kar[n, ki][m0 + 1].key < kar[m0 / 2].key, m0 < # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ n = 0, ki.key < kar[m1 / 2].key

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [n, m1]

Additional information

Used simplifier rules: ⊦ m + 1 ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ ¬ m < 0; (s)%n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n, m1