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Proof step 36 from proof prepost-remove (previous step)

kar[m1] = key-index(key0, m0), ar1[m] = elemindex(a, m2 + 1), kar[m2 + 1] = key-index(key, m), kar[m2 / 2] = key-index(key1, m3), 2exp-size(# kar), m2 / 2 < # kar, key0 < key1, m2 + 1 < # kar, m1 < # kar, m0 < # ar1, m2 < m1, m2 < # kar, m2 + 1 < m1, 2 + m2 < # kar, heap-order(kar, m1 + 1), no-dups-idx(kar, m1 + 1), valid-idx(kar, ar1, m1 + 1), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m1) ⊦ #(keyset(kar, m1) ++ key-index(key0, m0)) = 0, # kar = 1, m1 = m2 + 1, heap-order(kar[m2 + 1, key-index(key1, m3)], m1)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: bin-min-heap-inv name: heap-order-pull-down-parent seq: ⊦ heap-order(kar, n) ∧ m < n ∧ m ≠ 0 ∧ n ≤ # kar ∧ kar[(m - 1) / 2] = ki → heap-order(kar[m, ki], n) subst: [n, ki, m, kar] → [m1, key-index(key1, m3), m2 + 1, kar] flags: <all paths>

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%n < # kar ⊦ heap-order(kar, n + 1) → heap-order(kar, n); (ss)% ⊦ m + 1 ≠ 0; (s)% ⊦ m < n → m ≤ n; (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ n + 0 = n; (s) Used terms: m1, key-index(key1, m3), m2 + 1, kar