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Proof step 18 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, n ∈ nx, n ≤ m3, n < m3 → gc-heap(lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)], nx, m3) = gc-heap(lpt, nx, m3), m3 ∈ nx ∨ lpf ≠ LP_GROUP_NODES ⊦ gc-heap(lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)], nx, m3) = gc-heap(lpt, nx, m3) ∪ (¬ m3 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m3);∅)

Heuristic if-then-else split applied rule case distinction

And closed the goal!

Rule argument

pos: right 1

Additional information

Used simplifier rules: gc-heap-write-above: ⊦ n < # lpt ∧ m ≤ n → gc-heap(lpt[n, lp], nx, m) = gc-heap(lpt, nx, m);% ⊦ 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]% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m ≤ n → (n ≤ m ↔ m = n); (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ (gchs ++ ki) ∪ gchs0 = (gchs ∪ gchs0) ++ ki; (s)