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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

Rule argument

Additional information

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