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Proof step 32 from proof prepost-resize (previous step)

kar[m1] = key-index(key, m2), kar[m0] = key-index(key0, m2), 2exp-size(# kar), 2exp-size(n0), m0 < # kar, m0 < minsize, m1 < m, m < n0, m < minsize, m0 < m, m1 + 1 < minsize, m1 < # kar, m1 < minsize, m0 + 1 < minsize, m1 + 1 < n0, m0 + 1 < n0, m1 < n0, m0 < n0, heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar, minsize ≤ n0 ⊦ m1 = m0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 33:
⊦ no-dups-idx(kar, n) ↔ (∀ m0, m1. m0 < # kar ∧ m0 < n ∧ m1 < # kar ∧ m1 < n ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx)
Proof step 34:
kar[m0] = key-index(key0, m2), kar[m1] = key-index(key, m2), 2exp-size(n0), 2exp-size(# kar), m0 < n0, m1 < n0, m0 + 1 < n0, m1 + 1 < n0, m0 + 1 < minsize, m1 < minsize, m1 < # kar, m1 + 1 < minsize, m0 < m, m < minsize, m < n0, m1 < m, m0 < minsize, m0 < # kar, heap-order(kar, m), valid-idx(kar, ar, m), minsize ≤ n0, m ≤ # kar, ∀ m0, m1. m0 < # kar ∧ m0 < m ∧ m1 < # kar ∧ m1 < m ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx ⊦ m1 = m0

Rule argument

spec: bin-min-heap-inv name: no-dups-idx seq: ⊦ no-dups-idx(kar, n) ↔ (∀ m0, m1. m0 < # kar ∧ m0 < n ∧ m1 < # kar ∧ m1 < n ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx) subst: [n, kar] → [m, kar] flags: <all paths>

Additional information

Used terms: m, kar