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Proof step 45 from proof prepost-extract-min (previous step)

kar[0] = key-index(key1, m1), kar[m0] = key-index(key0, m), kar[1] = key-index(key2, m2), kar[2] = key-index(key3, m3), ar = ar1[m, ar1[m].idx:= 0], bh0 = bin-heap(kar[0, key-index(key0, m)], m0), 2exp-size(# kar), m < # ar1, m0 < # kar, heap-order(kar, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar, ar1, m0 + 1), (m0 ≠ 1 → ¬ key2 < key0) ∧ (2 < m0 → ¬ key3 < key0) → heap-order(kar[0, key-index(key0, m)], m0), m0 ≠ 1 ∧ key2 < key0 ∨ 2 < m0 ∧ key3 < key0 → heap-order(kar[0, (2 < m0 ∧ key3 < key2 ⊃ key-index(key3, m3);key-index(key2, m2))], m0), ∀ 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) ⊦ #(keyset(kar, m0) ++ key-index(key0, m)) = 0, m0 = 0, ∀ bh0, ar. bh0.size = m0 ∧ # bh0.ar = # kar ∧ id-idx(ar1, ar) ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar, bh0.size) ∧ heap-order(bh0.ar, bh0.size) ∧ keyset(kar[0, key-index(key0, m)], bh0.size) = keyset(bh0.ar, bh0.size) → bh0.size ≤ # bh0.ar ∧ 2exp-size(# bh0.ar) ∧ (∃ key. (key-index(key, m1) = kar[bh0.size] ∨ key-index(key, m1) ∈ keyset(kar, bh0.size)) ∧ keyset(bh0.ar, bh0.size) = (keyset(kar, bh0.size) ++ kar[bh0.size]) -- key-index(key, m1) ∧ (∀ ki. ki = kar[bh0.size] ∨ ki ∈ keyset(kar, bh0.size) → ¬ ki.key < key))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 46:
# bh.ar = # kar, kar[0] = key-index(key1, m1), kar[bh.size] = key-index(key0, m), kar[1] = key-index(key2, m2), kar[2] = key-index(key3, m3), keyset(kar[0, key-index(key0, m)], bh.size) = keyset(bh.ar, bh.size), 2exp-size(# kar), m < # ar1, bh.size < # kar, heap-order(kar, bh.size + 1), heap-order(bh.ar, bh.size), id-idx(ar1, ar0), no-dups-idx(kar, bh.size + 1), no-dups-idx(bh.ar, bh.size), valid-idx(kar, ar1, bh.size + 1), valid-idx(bh.ar, ar0, bh.size), (bh.size ≠ 1 → ¬ key2 < key0) ∧ (2 < bh.size → ¬ key3 < key0) → heap-order(kar[0, key-index(key0, m)], bh.size), bh.size ≠ 1 ∧ key2 < key0 ∨ 2 < bh.size ∧ key3 < key0 → heap-order(kar[0, (2 < bh.size ∧ key3 < key2 ⊃ key-index(key3, m3);key-index(key2, m2))], bh.size), ∀ ki. ki = key-index(key0, m) ∨ ki ∈ keyset(kar, bh.size) → ¬ ki.key < key, key-index(key, m1) = key-index(key0, m) ∨ key-index(key, m1) ∈ keyset(kar, bh.size) ⊦ #(keyset(kar, bh.size) ++ key-index(key0, m)) = 0, bh.size = 0, ∃ key. (key-index(key, m1) = key-index(key0, m) ∨ key-index(key, m1) ∈ keyset(kar, bh.size)) ∧ keyset(bh.ar, bh.size) = (keyset(kar, bh.size) ++ key-index(key0, m)) -- key-index(key, m1) ∧ (∀ ki. ki = key-index(key0, m) ∨ ki ∈ keyset(kar, bh.size) → ¬ ki.key < key)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s)% ⊦ ∅ ≠ s ++ ki; (s)%size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic[keyindex-set]%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic