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Proof step 49 from proof prepost-remove (previous step)

# kar0 = # kar, (keyset(kar, m1) -- key-index(key, m)) ++ key-index(key0, m0) = keyset(kar0, m1), 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), 2exp-size(# kar), m2 + 1 < m1, 2 + m2 < # kar, 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), heap-order(kar0, m1), id-idx(ar1, ar0), no-dups-idx(kar, m1 + 1), no-dups-idx(kar0, m1), valid-idx(kar, ar1, m1 + 1), valid-idx(kar0, ar0, m1), elemindex(a, m2 + 1) = elemindex(a, m0 + 1) → kar[m0 + 1] = key-index(key, m), 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, (keyset(kar, m1) -- key-index(key, m)) ++ key-index(key0, m0) = (keyset(kar, m1) ++ key-index(key0, m0)) -- key-index(key, m), m1 = m2 + 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 50:
⊦ ki ≠ ki0 → (s ++ ki0) -- ki = (s -- ki) ++ ki0
Proof step 51:
# kar0 = # kar, (keyset(kar, m1) -- key-index(key, m)) ++ key-index(key0, m0) = keyset(kar0, m1), 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), key-index(key, m) = key-index(key0, m0), 2exp-size(# kar), m2 / 2 < # kar, key0 < key1, m2 + 1 < # kar, m1 < # kar, m0 < # ar1, m2 < m1, m2 < # kar, 2 + m2 < # kar, m2 + 1 < m1, heap-order(kar0, m1), heap-order(kar, m1 + 1), heap-order(kar[m2 + 1, key-index(key1, m3)], m1), id-idx(ar1, ar0), no-dups-idx(kar0, m1), no-dups-idx(kar, m1 + 1), valid-idx(kar0, ar0, m1), valid-idx(kar, ar1, m1 + 1), elemindex(a, m2 + 1) = elemindex(a, m0 + 1) → kar[m0 + 1] = key-index(key, m), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m1) ⊦ # kar = 1, #(keyset(kar, m1) ++ key-index(key0, m0)) = 0, (keyset(kar, m1) -- key-index(key, m)) ++ key-index(key0, m0) = (keyset(kar, m1) ++ key-index(key0, m0)) -- key-index(key, m), m1 = m2 + 1

Rule argument

spec: set-union, inst: keyindex-set name: delsw seq: ⊦ ki ≠ ki0 → (s ++ ki0) -- ki = (s -- ki) ++ ki0 subst: [ki, ki0, s] → [key-index(key, m), key-index(key0, m0), keyset(kar, m1)] flags: <all paths>

Additional information

Used terms: key-index(key, m), key-index(key0, m0), keyset(kar, m1)