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



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Proof step 35 from proof prepost-remove (previous step)

kar[m1] = key-index(key0, m0), ar1[m] = elemindex(a, m2 + 1), kar[m2 + 1] = key-index(key, m), kar[m2 / 2] = key-index(key1, m3), ar = ar1[m0, ar1[m0].idx:= (m2 + 1)], bh = bin-heap(kar[m2 + 1, key-index(key0, m0)], m1), 2exp-size(# kar), m2 / 2 < # kar, key0 < key1, m2 + 1 < # kar, m1 < # kar, m0 < # 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) = 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 = bh ∧ ar = ar ∧ bubble-up-inv(m2 + 1, bh, ar, bh, ar)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: bubble-up-inv: ⊦ bubble-up-inv(n, bh, ar, bh0, ar0) ↔ n < bh.size ∧ bh.size ≤ # bh.ar ∧ # bh.ar = # bh0.ar ∧ bh.size = bh0.size ∧ id-idx(ar0, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ ( n = 0 ∨ ¬ bh.ar[n].key < bh.ar[(n - 1) / 2].key → heap-order(bh.ar, bh.size)) ∧ ( n ≠ 0 ∧ bh.ar[n].key < bh.ar[(n - 1) / 2].key → heap-order(bh.ar[n, bh.ar[(n - 1) / 2]], bh.size)) ∧ keyset(bh.ar, bh.size) = keyset(bh0.ar, bh0.size);%m < # kar ⊦ kar[m, ki][m, ki0] = kar[m, ki0]; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat% ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ m / 2 ≠ m + 1; (s)% ⊦ n + 0 = n; (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ key-index(key, m).key = key; (s)% ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ m + 1 ≠ 0; (s)%⊦ m < # kar ∧ valid-idx(kar, ar, m + 1) ∧ kar[m] = key-index(key, m1) → valid-idx(kar[m0, key-index(key, m1)], ar[m1, ar[m1].idx:= m0], m); (s)%⊦ m < # kar ∧ no-dups-idx(kar, m + 1) ∧ kar[m] = ki → no-dups-idx(kar[m0, ki], m); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ id-idx(ar, ar); (s)% ⊦ id-idx(ar1, ar0[n, ar0[n].idx:= m]) ↔ id-idx(ar1, ar0); (s)% ⊦ id-idx(ar1[m, ar1[m].idx:= m0], ar0) ↔ id-idx(ar1, ar0); (s)% ⊦ m < n → m ≤ n; (s)% ⊦ # kar[n, ki] = # kar; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)% ⊦ bin-heap(kar, m).size = m; (s)