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Proof step 11 from proof gc-heap-write-other-h (previous step)

lpt[n, mklp(n0, n1, lpf1, n2)][m3] = mklp(m6, m7, lpf2, m8), lpt[m3] = mklp(m, m4, lpf, m5), lpt[n] = mklp(m0, m1, lpf0, m2), n < # lpt, m3 < # lpt, lpt[n] = mklp(m0, m1, lpf0, m2) ∧ n < # lpt ∧ m3 ≤ # lpt → gc-heap(lpt[n, mklp(n0, n1, lpf1, n2)], nx, m3) = gc-heap(lpt, nx, m3) ∨ lpf0 = LP_GROUP_NODES ∨ lpf1 = LP_GROUP_NODES ⊦ lpf0 = LP_GROUP_NODES, lpf1 = LP_GROUP_NODES, gc-heap(lpt[n, mklp(n0, n1, lpf1, n2)], nx, m3) ∪ (¬ m3 ∈ nx ∧ lpf2 = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m3);∅) = gc-heap(lpt, nx, m3) ∪ (¬ m3 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m3);∅)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
lpt[n] = mklp(m0, m1, lpf0, m2), lpt[m3] = mklp(m, m4, lpf, m5), lpt[n, mklp(n0, n1, lpf1, n2)][m3] = mklp(m6, m7, lpf2, m8), gc-heap(lpt[n, mklp(n0, n1, lpf1, n2)], nx, m3) = gc-heap(lpt, nx, m3), m3 < # lpt, n < # lpt ⊦ lpf0 = LP_GROUP_NODES, lpf1 = LP_GROUP_NODES, gc-heap(lpt, nx, m3) ∪ (¬ m3 ∈ nx ∧ lpf2 = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m3);∅) = gc-heap(lpt, nx, m3) ∪ (¬ m3 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m3);∅)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s)