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Proof step 36 from proof bin-heap-inv-write-other (previous step)

kar[m9] = key-index(natkey, n), lpt[n] = mklp(m4, m5, lpf, m6), 2exp-size(# kar), m9 < m, m9 < # kar, n < # lpt, heap-order(kar, m), no-dups-idx(kar, m), m ≤ # kar ⊦ lpf = LP_GROUP_NODES, m6 = m9, key-index(natkey, n) ∈ keyset(kar, m)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 37:
n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m])
Proof step 38:
lpt[n] = mklp(m4, m5, lpf, m6), kar[m9] = key-index(natkey, n), 2exp-size(# kar), n < # lpt, m9 < # kar, m9 < m, heap-order(kar, m), no-dups-idx(kar, m), m ≤ # kar ⊦ lpf = LP_GROUP_NODES, m6 = m9, ∃ m0. m0 < m ∧ key-index(natkey, n) = kar[m0]

Rule argument

spec: bin-min-heap-inv, inst: gc-heap-rename-gc-heap-base 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(natkey, n)] flags: <all paths>

Additional information

Used terms: m, kar, key-index(natkey, n)