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Proof step 56 from proof prepost-resize (previous step)

copy(kar, 0, mkkeyarray(n0), 0, m)[m1 + 1] = key-index(key, m0), copy(kar, 0, mkkeyarray(n0), 0, m)[m1 / 2] = key-index(key0, m2), 2exp-size(# kar), 2exp-size(n0), m1 + 1 < # kar, m1 + 1 < minsize, m < n0, m < minsize, m1 + 1 < m, key < key0, m1 + 1 < n0, 2 + m1 < n0, 2 + m1 < minsize, heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar, minsize ≤ n0 ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 57:
⊦ m0 < # kar → copy(kar0, n, kar, n0, m)[m0] = (m0 < n0 ∨ n0 + m ≤ m0 ⊃ kar[m0];kar0[m0 + n - n0])
Proof step 58:
(m1 + 1 < 0 ∨ 0 + m ≤ m1 + 1 ⊃ mkkeyarray(n0)[m1 + 1];kar[(m1 + 1) + 0 - 0]) = key-index(key, m0), copy(kar, 0, mkkeyarray(n0), 0, m)[m1 / 2] = key-index(key0, m2), 2exp-size(n0), 2exp-size(# kar), 2 + m1 < minsize, 2 + m1 < n0, m1 + 1 < n0, key < key0, m1 + 1 < m, m < minsize, m < n0, m1 + 1 < minsize, m1 + 1 < # kar, m1 + 1 < # mkkeyarray(n0), heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), minsize ≤ n0, m ≤ # kar ⊦

Rule argument

spec: array-copy, inst: key-index-array name: at-copy seq: ⊦ m0 < # kar → copy(kar0, n, kar, n0, m)[m0] = (m0 < n0 ∨ n0 + m ≤ m0 ⊃ kar[m0];kar0[m0 + n - n0]) subst: [m0, m, n0, kar, n, kar0] → [m1 + 1, m, 0, mkkeyarray(n0), 0, kar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # mkkeyarray(n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%# kar = m, # kar0 = m ⊦ copy(kar, 0, kar0, 0, m) = kar; (s) Used terms: m1 + 1, m, 0, mkkeyarray(n0), 0, kar