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

lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), lpt[m3] = mklp(m, m4, lpf, m5), n < # lpt, m3 < # lpt, n ≤ m3, ∀ lpt, n, n0, n1, n2, m0, m1, m2, nx. lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2) ∧ n < m3 ∧ n < # lpt ∧ m3 ≤ # lpt → 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 ∈ nx ⊦ gc-heap(lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)], nx, m3) ∪ (¬ m3 ∈ nx ∧ lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)][m3].flags = LP_GROUP_NODES ⊃ ∅ ++ key-index(lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)][m3].ref-size, 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 elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 10:
lpt[m3] = mklp(m, m4, lpf, m5), lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)][m3] = mklp(m6, m7, lpf0, m8), m3 < # lpt, n < # lpt, n ≤ m3, ∀ lpt, n, n0, n1, n2, m0, m1, m2, nx. lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2) ∧ n < m3 ∧ n < # lpt ∧ m3 ≤ # lpt → 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 ∈ nx ⊦ 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

Rule argument

spec: lprops name: elim seq: ⊦ ((m = lp.ref-size ∧ m0 = lp.size) ∧ lpf = lp.flags) ∧ m1 = lp.gcheapidx ↔ lp = mklp(m, m0, lpf, m1) subst: [lp, m, m0, lpf, m1] → [lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)][m3], m6, m7, lpf0, m8]

Additional information

Used simplifier rules: ⊦ ((m = lp.ref-size ∧ m0 = lp.size) ∧ lpf = lp.flags) ∧ m1 = lp.gcheapidx ↔ lp = mklp(m, m0, lpf, m1); (elim) Used terms: lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)][m3], m6, m7, lpf0, m8