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Proof step 13 from proof gc-heap-free-h (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

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);%size-zero-01: ⊦ gchs ≠ ∅ → # gchs ≠ 0; (forward) from specification set-basic[gc-heap-rename-gc-heap-base-keyindex-set]%ne-01: ⊦ ki ∈ gchs → gchs ≠ ∅; (forward) from specification set-basic[gc-heap-rename-gc-heap-base-keyindex-set]% ⊦ gchs -- ki = gchs ↔ ¬ ki ∈ gchs; (s)% ⊦ ∅ -- ki = ∅; (s)% ⊦ (gchs ++ ki) -- ki = gchs -- ki; (s)%unc: ⊦ gchs ∪ gchs0 = gchs0 ∪ gchs; from specification set-union[gc-heap-rename-gc-heap-base-keyindex-set]%una: ⊦ (gchs ∪ gchs0) ∪ gchs1 = gchs ∪ gchs0 ∪ gchs1; from specification set-union[gc-heap-rename-gc-heap-base-keyindex-set]% ⊦ gchs ∪ ∅ = gchs; (s)% ⊦ LP_FREE ≠ LP_GROUP_NODES; (s)% ⊦ m ≤ m; (s)%⊦ mklp(m, m0, lpf, m1) = mklp(m2, m3, lpf0, m4) ↔ m = m2 ∧ m0 = m3 ∧ lpf = lpf0 ∧ m1 = m4; (ws)% ⊦ m < # lpt → lpt[m, lp][m] = lp; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ m ≤ n → (n ≤ m ↔ m = n); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)