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Proof step 20 from proof gc-heap-update-h (previous step)

lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), lpt[m3] = mklp(m, m4, lpf, m5), 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 + 1 < # lpt, m3 < # lpt, n < # lpt, n < m3 ⊦ ((gc-heap(lpt, nx, m3) -- key-index(n0, n)) ∪ (¬ m3 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m3);∅)) ++ key-index(m0, n) = ((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), m3 = n, n ∈ nx

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Proof step 21:
lpt[m3] = mklp(m, m4, lpf, m5), lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), 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), lpf = LP_GROUP_NODES, n < m3, n < # lpt, m3 < # lpt, n + 1 < # lpt ⊦ ((gc-heap(lpt, nx, m3) -- key-index(n0, n)) ∪ (∅ ++ key-index(m, m3))) ++ key-index(m0, n) = ((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), m3 = n, n ∈ nx, m3 ∈ nx

Rule argument

pos: right 1

Additional information

Used simplifier rules: ⊦ gchs ∪ ∅ = gchs; (s)%una: ⊦ (gchs ∪ gchs0) ∪ gchs1 = gchs ∪ gchs0 ∪ gchs1; from specification set-union[gc-heap-rename-gc-heap-base-keyindex-set]%unc: ⊦ gchs ∪ gchs0 = gchs0 ∪ gchs; from specification set-union[gc-heap-rename-gc-heap-base-keyindex-set]% ⊦ ∅ -- ki = ∅; (s)