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Proof step 9 from proof gc-heap-in-h (previous step)

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

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Proof step 10:
key-index(n0, n1) ∈ gc-heap(lpt, nx, n2) ↔ n1 < n2 ∧ ¬ n1 ∈ nx ∧ m = n0 ∧ lpf = LP_GROUP_NODES, lpt[n2] = mklp(m2, m3, lpf0, m4), lpt[n1] = mklp(m, m0, lpf, m1), n2 < # lpt ⊦ key-index(n0, n1) ∈ gc-heap(lpt, nx, n2) ∪ (¬ n2 ∈ nx ∧ lpf0 = LP_GROUP_NODES ⊃ ∅ ++ key-index(m2, n2);∅) ↔ ¬ n2 < n1 ∧ ¬ n1 ∈ nx ∧ m = n0 ∧ lpf = LP_GROUP_NODES, key-index(n0, n1) ∈ gc-heap(lpt, nx, n2)

Rule argument

fma: key-index(n0, n1) ∈ gc-heap(lpt, nx, n2)

Additional information

Used simplifier rules: ⊦ m < n → ¬ n < m; (s)% ⊦ n < n0 → (n0 < n ↔ false); (ss)% ⊦ ki ∈ gchs ∪ gchs0 ↔ ¬(¬ ki ∈ gchs ∧ ¬ ki ∈ gchs0); (ss)%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] Used formulas: key-index(n0, n1) ∈ gc-heap(lpt, nx, n2)