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

lpt[m3] = mklp(m, m4, lpf, m5), lpt[n] = mklp(m0, m1, LP_FREE, m2), n < # lpt, m3 < # lpt, m3 < # lpt[n, lp], ∀ lpt, n, m0, m1, m2, lp, nx. lpt[n] = mklp(m0, m1, LP_FREE, m2) ∧ n < # lpt ∧ m3 ≤ # lpt → gc-heap(lpt[n, lp], nx + n ', m3) = gc-heap(lpt, nx, m3) ⊦ gc-heap(lpt[n, lp], nx + n ', m3) ∪ (¬ m3 ∈ nx + n ' ∧ lpt[n, lp][m3].flags = LP_GROUP_NODES ⊃ {key-index(lpt[n, lp][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(m0, m1, LP_FREE, m2), n < # lpt, m3 < # lpt, ∀ lpt, n, m0, m1, m2, lp, nx. lpt[n] = mklp(m0, m1, LP_FREE, m2) ∧ n < # lpt ∧ m3 ≤ # lpt → gc-heap(lpt[n, lp], nx + n ', m3) = gc-heap(lpt, nx, m3) ⊦ gc-heap(lpt[n, lp], nx + n ', m3) ∪ (¬ m3 ∈ nx ∧ m3 ≠ n ∧ 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[n, lp][m] = lpt[m]; (s)%a: ⊦ (nx + nx0) + nx1 = nx + nx0 + nx1; from specification list[nat-list-set]% ⊦ n ∈ n0 ' ↔ n = n0; (s)% ⊦ n ∈ nx + nx0 ↔ n ∈ nx ∨ n ∈ nx0; (s)