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Proof step 19 from proof gc-heap-update-h (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
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

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (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 ++ ki) ∪ gchs0 = (gchs ∪ gchs0) ++ ki; (s)% ⊦ m < n → n ≠ 0; (s)%⊦ mklp(m, m0, lpf, m1) = mklp(m2, m3, lpf0, m4) ↔ m = m2 ∧ m0 = m3 ∧ lpf = lpf0 ∧ m1 = m4; (ws)%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ m ≠ n → lpt[n, lp][m] = lpt[m]; (s)