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Proof step 18 from proof prepost-extract-min (previous step)

ar = ar1[bh0.ar[0].idx, ar1[bh0.ar[0].idx].idx:= 0], ar0 = ar1, bh = bin-heap(kar, m), bh0 = bh1.ar:= bh1.ar[0, bh1.ar[bh1.size]], bh1 = bin-heap(kar, m - 1), 2exp-size(# kar), bh1.size < # bh1.ar, 0 < # bh1.ar, 0 < # bh.ar, bh0.ar[0].idx < # ar1, 0 < # bh0.ar, heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar1, m), m ≤ # kar, m ≥ 1, ∃ key. key-index(key, bh.ar[0].idx) ∈ keyset(kar, m) ∧ (∀ ki. ki ∈ keyset(kar, m) → ¬ ki.key < key) ⊦ ⟪bin_min_heap_bubble_down#(0; bh0, ar)⟫ ( bh0.size ≤ # bh0.ar ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar, bh0.size) ∧ 2exp-size(# bh0.ar) ∧ heap-order(bh0.ar, bh0.size) ∧ id-idx(ar0, ar) ∧ (∃ key. key-index(key, bh.ar[0].idx) ∈ keyset(kar, m) ∧ keyset(bh0.ar, bh0.size) = keyset(kar, m) -- key-index(key, bh.ar[0].idx) ∧ (∀ ki. ki ∈ keyset(kar, m) → ¬ ki.key < key))), # keyset(kar, m) = 0, bh1.size = 0, keyset(kar, m) = ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
ar = ar1[kar[m - 1].idx, ar1[kar[m - 1].idx].idx:= 0], bh0 = bin-heap(kar[0, kar[m - 1]], m - 1), # kar ≠ 0, # keyset(kar, m) ≠ 0, keyset(kar, m) ≠ ∅, m ≠ 0, m ≠ 1, 2exp-size(# kar), kar[m - 1].idx < # ar1, heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar1, m), key-index(key, kar[0].idx) ∈ keyset(kar, m), m ≤ # kar, ∀ ki. ki ∈ keyset(kar, m) → ¬ ki.key < key ⊦ ⟪bin_min_heap_bubble_down#(0; bh0, ar)⟫ ( bh0.size ≤ # bh0.ar ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar, bh0.size) ∧ 2exp-size(# bh0.ar) ∧ heap-order(bh0.ar, bh0.size) ∧ id-idx(ar1, ar) ∧ (∃ key0. key-index(key0, kar[0].idx) ∈ keyset(kar, m) ∧ keyset(bh0.ar, bh0.size) = keyset(kar, m) -- key-index(key0, kar[0].idx) ∧ (∀ ki. ki ∈ keyset(kar, m) → ¬ ki.key < key0)))

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (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% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ keyset(kar, 0) = ∅; (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)%m < # kar ⊦ kar[m, ki][m] = ki; (s)% ⊦ # kar[n, ki] = # kar; (s)% ⊦ m ≤ n → (n - m = 0 ↔ n = m); (s)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ 0 < m ↔ m ≠ 0; (s)% ⊦ bin-heap(kar, m).ar:= kar0 = bin-heap(kar0, m); (s)% ⊦ bin-heap(kar, m).size = m; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)