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Proof step 37 from proof prepost-extract-min (previous step)

kar[1] = key-index(key3, m3), kar[2] = key-index(key2, m2), kar[0] = key-index(key1, m1), kar[m0] = key-index(key0, m), 2exp-size(# kar), m < # ar1, m0 < # kar, heap-order(kar, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar, ar1, m0 + 1), ∀ ki. ki = key-index(key0, m) ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key, key-index(key, m1) = key-index(key0, m) ∨ key-index(key, m1) ∈ keyset(kar, m0), m0 ≠ 1 ∧ key3 < key0 ∨ 2 < m0 ∧ key2 < key0 ⊦ #(keyset(kar, m0) ++ key-index(key0, m)) = 0, m0 = 0, heap-order(kar[0, (2 < m0 ∧ key2 < key3 ⊃ key-index(key2, m2);key-index(key3, m3))], m0)

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] → [m0, (2 < m0 ∧ key2 < key3 ⊃ key-index(key2, m2) ;key-index(key3, m3) ), 0, kar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s)% ⊦ 0 < m ↔ m ≠ 0; (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)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)% ⊦ key-index(key, m).key = key; (s)% ⊦ m < n → n ≠ 0; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < 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]% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)%ne-01: ⊦ ki ∈ s → s ≠ ∅; (forward) from specification set-basic[keyindex-set] Used terms: m0, (2 < m0 ∧ key2 < key3 ⊃ key-index(key2, m2);key-index(key3, m3)), 0, kar