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Proof step 6 from proof gc-heap-write-above (previous step)

lpt[m0] = mklp(m, m1, lpf, m2), m0 + 1 < # lpt, n < # lpt, m0 < n, m0 < # lpt, ∀ lpt, n, lp, nx. m0 < # lpt ∧ n < # lpt ∧ m0 ≤ n → gc-heap(lpt[n, lp], nx, m0) = gc-heap(lpt, nx, m0) ⊦ gc-heap(lpt[n, lp], nx, m0 + 1) = gc-heap(lpt, nx, m0) ∪ (¬ m0 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m0);∅)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ 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 8:
lpt[m0] = mklp(m, m1, lpf, m2), m0 < # lpt, m0 < n, n < # lpt, m0 + 1 < # lpt, m0 < # lpt[n, lp], ∀ lpt, n, lp, nx. m0 < # lpt ∧ n < # lpt ∧ m0 ≤ n → gc-heap(lpt[n, lp], nx, m0) = gc-heap(lpt, nx, m0) ⊦ gc-heap(lpt[n, lp], nx, m0) ∪ (¬ m0 ∈ nx ∧ lpt[n, lp][m0].flags = LP_GROUP_NODES ⊃ {key-index(lpt[n, lp][m0].ref-size, m0)} ;∅) = gc-heap(lpt, nx, m0) ∪ (¬ m0 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m0);∅)

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] → [m0, nx, lpt[n, lp]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # lpt[n, lp] = # lpt; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: m0, nx, lpt[n, lp]