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Proof step 43 from proof prepost-bubble-up (previous step)

# kar0 = # kar, kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)][(m / 2 - 1) / 2] = key-index(key3, m5), kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), kar0[m / 2] = key-index(key1, m3), kar0[m + 1] = key-index(key2, m4), keyset(kar0, m2) = keyset(kar, m2), m + 1 < # kar, m + 1 < m2, n1 < # kar, n1 < m2, m3 < # ar, key2 < key1, m4 < # ar, m / 2 < m2, m / 2 < # kar, heap-order(kar0[m + 1, key-index(key1, m3)], m2), id-idx(ar1, ar), no-dups-idx(kar, m2), no-dups-idx(kar0, m2), valid-idx(kar, ar1, m2), valid-idx(kar0, ar, m2), m2 ≤ # kar, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), m = 0 ∨ m = 1 ∨ ¬ key2 < key3 ⊦ # kar = 1, m2 = 1, heap-order(kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)], m2)

User applied rule cut formula

And obtained the following new goals

Proof step 44:
# kar0 = # kar, kar0[m + 1] = key-index(key2, m4), kar0[m / 2] = key-index(key1, m3), kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)][(m / 2 - 1) / 2] = key-index(key3, m5), keyset(kar0, m2) = keyset(kar, m2), m = 0, m / 2 < # kar, m / 2 < m2, m4 < # ar, key2 < key1, m3 < # ar, n1 < m2, n1 < # kar, m + 1 < m2, m + 1 < # kar, heap-order(kar0[m + 1, key-index(key1, m3)], m2), id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar0, ar, m2), valid-idx(kar, ar1, m2), m2 ≤ # kar, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), m = 0 ∨ m = 1 ∨ ¬ key2 < key3 ⊦ # kar = 1, m2 = 1, heap-order(kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)], m2)
Proof step 47:
# kar0 = # kar, kar0[m + 1] = key-index(key2, m4), kar0[m / 2] = key-index(key1, m3), kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)][(m / 2 - 1) / 2] = key-index(key3, m5), keyset(kar0, m2) = keyset(kar, m2), m / 2 < # kar, m / 2 < m2, m4 < # ar, key2 < key1, m3 < # ar, n1 < m2, n1 < # kar, m + 1 < m2, m + 1 < # kar, heap-order(kar0[m + 1, key-index(key1, m3)], m2), id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar0, ar, m2), valid-idx(kar, ar1, m2), m2 ≤ # kar, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), m = 0 ∨ m = 1 ∨ ¬ key2 < key3 ⊦ # kar = 1, m = 0, m2 = 1, heap-order(kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)], m2)

Rule argument

fma: m = 0

Additional information

Used formulas: m = 0