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Proof step 79 from proof prepost-remove (previous step)

kar[2 * m1 + 1] = key-index(key3, m5), kar[2 * m1 + 2] = key-index(key2, m4), kar[m1, key-index(key0, m0)][(m1 - 1) / 2] = key-index(key1, m3), ar1[m] = elemindex(a, m1), kar[m1] = key-index(key, m), kar[m2] = key-index(key0, m0), 2exp-size(# kar), m1 + 1 < # kar, m1 < # kar, m2 < # kar, m0 < # ar1, m1 < m2, heap-order(kar, m2 + 1), no-dups-idx(kar, m2 + 1), valid-idx(kar, ar1, m2 + 1), elemindex(a, m1) = elemindex(a, m3) ∧ valid-idx(kar, ar1, m1) ∧ key-index(key, m) ∈ keyset(kar, m1) → kar[m3] = key-index(key, m), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1, key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m2), 2 * m1 + 1 < m2 ∧ key3 < key0 ∨ 2 * m1 + 2 < m2 ∧ key2 < key0 ⊦ #(keyset(kar, m2) ++ key-index(key0, m0)) = 0, m2 = m1, heap-order(kar[m1, (2 * m1 + 2 < m2 ∧ key2 < key3 ⊃ key-index(key2, m4);key-index(key3, m5))], m2), m1 ≠ 0 ∧ key0 < key1

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: bin-min-heap-inv name: heap-order-pull-up-child seq: n ≤ # kar, m < n ⊦ heap-order(kar, n) ∧ 2 * m + 1 < n ∧ ki = ( 2 * m + 2 < n ∧ kar[2 * m + 2].key < kar[2 * m + 1].key ⊃ kar[2 * m + 2] ;kar[2 * m + 1] ) → (¬ heap-order(kar[m, ki], n) ↔ false) subst: [n, ki, m, kar] → [m2, (2 * m1 + 2 < m2 ∧ key2 < key3 ⊃ key-index(key2, m4) ;key-index(key3, m5) ), m1, kar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s)%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)% ⊦ key-index(key, m).key = key; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%nll: ⊦ ¬ key < key0 ∧ key < key1 → key0 < key1; (forward) from specification oelem[binheap-key]% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ 2 * n < n; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ m +1 = m + 1; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%ne-01: ⊦ ki ∈ s → s ≠ ∅; (forward) from specification set-basic[keyindex-set] Used terms: m2, (2 * m1 + 2 < m2 ∧ key2 < key3 ⊃ key-index(key2, m4) ;key-index(key3, m5) ), m1, kar