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Proof step 9 from proof heap-order-add-parent-key (previous step)

kar[m1 + 1, key-index(key, m)][m3 + 1] = key-index(key0, m0), kar[m1 / 2] = key-index(key, m), kar[m1 + 1, key-index(key, m)][m3 / 2] = key-index(key1, m4), m3 + 1 < # kar, key0 < key1, m1 + 1 < # kar, m3 ≤ m1, ∀ m2. ¬ m1 < m2 ∧ m2 ≠ 0 → ¬ kar[m2].key < kar[(m2 - 1) / 2].key ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ m ≠ n → kar[n, ki][m] = kar[m]
Proof step 11:
kar[m1 + 1, key-index(key, m)][m3 / 2] = key-index(key1, m4), kar[m1 / 2] = key-index(key, m), kar[m3 + 1] = key-index(key0, m0), m1 + 1 < # kar, key0 < key1, m3 + 1 < # kar, m3 ≤ m1, ∀ m2. ¬ m1 < m2 ∧ m2 ≠ 0 → ¬ kar[m2].key < kar[(m2 - 1) / 2].key ⊦ m3 + 1 = m1 + 1

Rule argument

spec: array, inst: key-index-array name: Put-other seq: ⊦ m ≠ n → kar[n, ki][m] = kar[m] subst: [m, ki, n, kar] → [m3 + 1, key-index(key, m), m1 + 1, kar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + n = m + n0 ↔ n = n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m / 2 ≠ m + 1; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)% ⊦ m ≤ m; (s)%lel: ⊦ key < key0 ∧ key = key1 → key1 < key0; (forward) from specification oelem[binheap-key]% ⊦ ¬ key < key; (s) Used terms: m3 + 1, key-index(key, m), m1 + 1, kar