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Proof step 40 from proof prepost-remove (previous step)

kar[m2 / 2] = key-index(key1, m3), kar[m2 + 1] = key-index(key, m), ar1[m] = elemindex(a, m2 + 1), kar[m1] = key-index(key0, m0), ar = ar1[m0, ar1[m0].idx:= (m2 + 1)], bh = bin-heap(kar[m2 + 1, key-index(key0, m0)], m1), 2exp-size(# kar), 2 + m2 < # kar, m2 + 1 < m1, m2 < # kar, m2 < m1, m0 < # ar1, m1 < # kar, m2 + 1 < # kar, key0 < key1, m2 / 2 < # kar, heap-order(kar[m2 + 1, key-index(key1, m3)], m1), heap-order(kar, m1 + 1), no-dups-idx(kar, m1 + 1), valid-idx(kar, ar1, m1 + 1), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m1) ⊦ #(keyset(kar, m1) ++ key-index(key0, m0)) = 0, # kar = 1, m1 = m2 + 1, ∀ bh, ar. bh.size = m1 ∧ # bh.ar = # kar ∧ id-idx(ar1, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ heap-order(bh.ar, bh.size) ∧ keyset(kar[m2 + 1, key-index(key0, m0)], bh.size) = keyset(bh.ar, bh.size) → bh.size ≤ # bh.ar ∧ 2exp-size(# bh.ar) ∧ keyset(bh.ar, bh.size) = (keyset(kar, bh.size) ++ kar[bh.size]) -- key-index(key, m)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 41:
# bh0.ar = # kar, kar[m2 / 2] = key-index(key1, m3), kar[m2 + 1] = key-index(key, m), ar1[m] = elemindex(a, m2 + 1), kar[bh0.size] = key-index(key0, m0), keyset(kar[m2 + 1, key-index(key0, m0)], bh0.size) = keyset(bh0.ar, bh0.size), 2exp-size(# kar), m2 + 1 < bh0.size, 2 + m2 < # kar, m2 < # kar, m2 < bh0.size, m0 < # ar1, bh0.size < # kar, m2 + 1 < # kar, key0 < key1, m2 / 2 < # kar, heap-order(kar[m2 + 1, key-index(key1, m3)], bh0.size), heap-order(kar, bh0.size + 1), heap-order(bh0.ar, bh0.size), id-idx(ar1, ar0), no-dups-idx(kar, bh0.size + 1), no-dups-idx(bh0.ar, bh0.size), valid-idx(kar, ar1, bh0.size + 1), valid-idx(bh0.ar, ar0, bh0.size), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, bh0.size) ⊦ #(keyset(kar, bh0.size) ++ key-index(key0, m0)) = 0, # kar = 1, bh0.size = m2 + 1, keyset(bh0.ar, bh0.size) = (keyset(kar, bh0.size) ++ key-index(key0, m0)) -- key-index(key, m)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s)% ⊦ ∅ ≠ s ++ ki; (s)% ⊦ m < n → n ≠ 0; (s)%size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic[keyindex-set]%succ-03: ⊦ m < n ∧ m + 1 ≠ n → m + 1 < n; (forward) from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic