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Proof step 8 from proof gc-heap-update-log-h (previous step)

lpt[m3] = mklp(m, m4, lpf, m5), lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), n < # lpt, m3 < # lpt, m3 < # lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)], n ∈ nx, n ≤ m3, ∀ lpt, n, n0, n1, n2, nx, m0, m1, m2. lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2) ∧ n < m3 ∧ n < # lpt ∧ n ∈ nx ∧ m3 ≤ # lpt → gc-heap(lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)], nx, m3) = gc-heap(lpt, nx, m3) ⊦ gc-heap(lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)], nx, m3) ∪ (¬ m3 ∈ nx ∧ lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)][m3].flags = LP_GROUP_NODES ⊃ {key-index(lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)][m3].ref-size, 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 9:
lpt[m3] = mklp(m, m4, lpf, m5), lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), n < # lpt, m3 < # lpt, n ∈ nx, n ≤ m3, ∀ lpt, n, n0, n1, n2, nx, m0, m1, m2. lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2) ∧ n < m3 ∧ n < # lpt ∧ n ∈ nx ∧ m3 ≤ # lpt → gc-heap(lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)], nx, m3) = gc-heap(lpt, nx, m3) ⊦ gc-heap(lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)], nx, m3) ∪ (¬ m3 ∈ nx ∧ lpf = 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: ⊦ mklp(n2, n3, lpf, m).ref-size = n2; (s)% ⊦ mklp(n2, n3, lpf, m).flags = lpf; (s)% ⊦ # lpt[n, lp] = # lpt; (s)% ⊦ {ki} = ∅ ++ ki; (s)%m ≠ n ⊦ lpt[m, lp][n] = lpt[n]; (s)