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Proof step 21 from proof prepost-get-min (previous step)

kar[0] = key-index(key0, m0), 2exp-size(# kar), key < key0, heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), key-index(key, m1) ∈ keyset(kar, m), m ≤ # kar ⊦ # kar = 0, # keyset(kar, m) = 0, keyset(kar, m) = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m])
Proof step 23:
kar[0] = key-index(key0, m0), 2exp-size(# kar), key < key0, heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar, ∃ m0. m0 < m ∧ key-index(key, m1) = kar[m0] ⊦ # keyset(kar, m) = 0, # kar = 0, keyset(kar, m) = ∅

Rule argument

spec: bin-min-heap-inv name: key-set-extract-index seq: n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m]) subst: [n, kar, ki] → [m, kar, key-index(key, m1)] flags: <all paths>

Additional information

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