[Imprint][Data Privacy Statement][E-Mail]

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Proof step 28 from proof prepost-remove (previous step)

kar[m2 + 1] = key-index(key, m), ar1[m] = elemindex(a, m2 + 1), ar = ar1[kar[m1].idx, ar1[kar[m1].idx].idx:= (m2 + 1)], bh = bin-heap(kar[m2 + 1, kar[m1]], m1), m0 = m1 + 1, 2exp-size(# kar), m2 + 1 < # kar, kar[m1].key < kar[m2 / 2].key, m2 / 2 < # kar, m2 + 1 < m0, kar[m1].idx < # ar1, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar1, m0), key-index(key, m) ∈ keyset(kar, m0), m0 ≤ # kar ⊦ ⟪bin_min_heap_bubble_up#(m2 + 1; bh, ar)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar1, ar) ∧ keyset(bh.ar, bh.size) = keyset(kar, m0) -- key-index(key, m)), # kar = 1, # keyset(kar, m0) = 0, keyset(kar, m0) = ∅, m0 = 0, m0 = 1, m1 = m2 + 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 29:
ar1[m] = elemindex(a, m2 + 1), kar[m2 + 1] = key-index(key, m), ar = ar1[kar[m1].idx, ar1[kar[m1].idx].idx:= (m2 + 1)], bh = bin-heap(kar[m2 + 1, kar[m1]], m1), # kar ≠ 1, #(keyset(kar, m1) ++ kar[m1]) ≠ 0, m1 ≠ m2 + 1, 2exp-size(# kar), m2 / 2 < # kar, kar[m1].key < kar[m2 / 2].key, m2 + 1 < # kar, m1 < # kar, kar[m1].idx < # ar1, m2 < m1, m2 < # kar, m2 + 1 < m1, 2 + m2 < # kar, heap-order(kar, m1 + 1), no-dups-idx(kar, m1 + 1), valid-idx(kar, ar1, m1 + 1), key-index(key, m) = kar[m1] ∨ key-index(key, m) ∈ keyset(kar, m1) ⊦ ⟪bin_min_heap_bubble_up#(m2 + 1; bh, ar)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar1, ar) ∧ keyset(bh.ar, bh.size) = (keyset(kar, m1) ++ kar[m1]) -- key-index(key, m))

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)%size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic[keyindex-set]%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%succ-03: ⊦ m < n ∧ m + 1 ≠ n → m + 1 < n; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ ki ∈ s ++ ki0 ↔ ki = ki0 ∨ ki ∈ s; (s)% ⊦ ∅ ≠ s ++ ki; (s)% ⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n]; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m + 1 ≠ 0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = 1 ↔ m = 0 ∧ n = 1 ∨ n = 0 ∧ m = 1; (ws)