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Proof step 11 from proof prepost-update (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m])
Proof step 13:
kar[m1] = key-index(key0, m), ar[m] = elemindex(a, m1), 2exp-size(# kar), heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), m0 ≤ m1, m0 ≤ # kar, ∃ m1. m1 < m0 ∧ key-index(key0, m) = kar[m1] ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅

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] → [m0, kar, key-index(key0, m)] flags: <all paths>

Additional information

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