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Proof step 22 from proof gc-heap-update-h (previous step)

lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), lpt[m3] = mklp(m, m4, LP_GROUP_NODES, 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)) ++ key-index(m, m3) ++ key-index(m0, n) = ((gc-heap(lpt, nx, m3) -- key-index(n0, n)) ∪ ((∅ ++ key-index(m, m3)) -- key-index(n0, n))) ++ key-index(m0, n), m3 = n, m3 ∈ nx, n ∈ nx

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ gchs = gchs0 ++ ki ↔ (∀ ki0. ki0 ∈ gchs ↔ ki0 = ki ∨ ki0 ∈ gchs0)
Proof step 24:
lpt[m3] = mklp(m, m4, LP_GROUP_NODES, m5), lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), 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 < m3, n < # lpt, m3 < # lpt, n + 1 < # lpt ⊦ m3 = n, n ∈ nx, m3 ∈ nx, ∀ ki0. ki0 ∈ (gc-heap(lpt, nx, m3) -- key-index(n0, n)) ++ key-index(m, m3) ++ key-index(m0, n) ↔ ki0 = key-index(m0, n) ∨ ki0 ∈ (gc-heap(lpt, nx, m3) -- key-index(n0, n)) ∪ ((∅ ++ key-index(m, m3)) -- key-index(n0, n))

Rule argument

spec: set-basic, inst: gc-heap-rename-gc-heap-base-keyindex-set name: ext-rec seq: ⊦ gchs = gchs0 ++ ki ↔ (∀ ki0. ki0 ∈ gchs ↔ ki0 = ki ∨ ki0 ∈ gchs0) subst: [ki, gchs0, gchs] → [key-index(m0, n), (gc-heap(lpt, nx, m3) -- key-index(n0, n)) ∪ ((∅ ++ key-index(m, m3)) -- key-index(n0, n)), (gc-heap(lpt, nx, m3) -- key-index(n0, n)) ++ key-index(m, m3) ++ key-index(m0, n)] flags: <all paths>

Additional information

Used terms: key-index(m0, n), (gc-heap(lpt, nx, m3) -- key-index(n0, n)) ∪ ((∅ ++ key-index(m, m3)) -- key-index(n0, n)), (gc-heap(lpt, nx, m3) -- key-index(n0, n)) ++ key-index(m, m3) ++ key-index(m0, n)