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Proof step 8 from proof gc-heap-remove-wbufs-h (previous step)

lpt[n0] = mklp(m, m0, lpf, m1), gc-heap(lpt.mapIndices(ns, λ lp. lp.size:= LEB_SIZE), nx, n0) = gc-heap(lpt, nx, n0), n0 < # lpt ⊦ gc-heap(lpt.mapIndices(ns, λ lp. lp.size:= LEB_SIZE), nx, n0 + 1) = gc-heap(lpt, nx, n0) ∪ (¬ n0 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, n0);∅)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ n < # lpt → gc-heap(lpt, nx, n + 1) = gc-heap(lpt, nx, n) ∪ (¬ n ∈ nx ∧ lpt[n].flags = LP_GROUP_NODES ⊃ {key-index(lpt[n].ref-size, n)} ;∅)
Proof step 10:
lpt[n0] = mklp(m, m0, lpf, m1), gc-heap(lpt.mapIndices(ns, λ lp. lp.size:= LEB_SIZE), nx, n0) = gc-heap(lpt, nx, n0), n0 < # lpt, n0 < # lpt.mapIndices(ns, λ lp. lp.size:= LEB_SIZE) ⊦ gc-heap(lpt.mapIndices(ns, λ lp. lp.size:= LEB_SIZE), nx, n0) ∪ (¬ n0 ∈ nx ∧ lpt.mapIndices(ns, λ lp. lp.size:= LEB_SIZE)[n0].flags = LP_GROUP_NODES ⊃ {key-index(lpt.mapIndices(ns, λ lp. lp.size:= LEB_SIZE)[n0].ref-size, n0)} ;∅ ) = gc-heap(lpt, nx, n0) ∪ (¬ n0 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, n0);∅)

Rule argument

spec: <toplevel> name: gc-heap-rec seq: ⊦ n < # lpt → gc-heap(lpt, nx, n + 1) = gc-heap(lpt, nx, n) ∪ (¬ n ∈ nx ∧ lpt[n].flags = LP_GROUP_NODES ⊃ {key-index(lpt[n].ref-size, n)} ;∅ ) subst: [n, nx, lpt] → [n0, nx, lpt.mapIndices(ns, λ lp. lp.size:= LEB_SIZE)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # lpt.mapIndices(ns, lptfun) = # lpt; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: n0, nx, lpt.mapIndices(ns, λ lp. lp.size:= LEB_SIZE)