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Proof step 22 from proof heap-order-overwrite-children-ok (previous step)

kar[2 * m + 2] = key-index(key3, m5), kar[m1 / 2] = key-index(key1, m3), kar[m, key-index(key, m0)][m1 + 1] = key-index(key0, m2), kar[m] = key-index(key2, m4), kar[2 * m + 1] = key-index(key4, m6), kar[m1 + 1] = key-index(key5, m7), m1 + 1 < # kar, m < # kar, m1 + 1 < n, key0 < key1, m < n, n ≤ # kar, 2 * m + 1 < n → ¬ key4 < key, 2 * m + 2 < n → ¬ key3 < key ⊦ m1 / 2 = m, key5 < key1, key < key2

User applied rule insert rewrite lemma

And obtained the following new goals

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] → [m1 + 1, key-index(key, m0), m, kar] flags: <all paths>

Additional information

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