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Proof step 3 from proof gc-heap-update-h (previous step)

lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), n < m, n < # lpt, m ≤ # lpt ⊦ gc-heap(lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)], nx, m) = (gc-heap(lpt, nx, m) -- key-index(n0, n)) ++ key-index(m0, n), n ∈ nx

User applied rule structural induction

And obtained the following new goals

Proof step 4:
lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), n < # lpt, n < m3 +1, m3 +1 ≤ # lpt, ∀ 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 +1) = (gc-heap(lpt, nx, m3 +1) -- key-index(n0, n)) ++ key-index(m0, n), n ∈ nx

Rule argument

indvar: m

Additional information

Used simplifier rules: gc-heap-base: ⊦ gc-heap(lpt, nx, 0) = ∅;%nat freely generated by 0, +1;;% ⊦ ∅ -- ki = ∅; (s)% ⊦ ∅ ≠ gchs ++ ki; (s)% ⊦ 0 ≤ m; (s)% ⊦ ¬ m < 0; (s)