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Proof step 5 from proof gc-heap-in-h (previous step)

lpt[n1] = mklp(m, m0, lpf, m1), n2 +1 ≤ # lpt, ∀ lpt, n1, m, m0, lpf, m1, n0, nx. lpt[n1] = mklp(m, m0, lpf, m1) ∧ n2 ≤ # lpt → (key-index(n0, n1) ∈ gc-heap(lpt, nx, n2) ↔ n1 < n2 ∧ ¬ n1 ∈ nx ∧ m = n0 ∧ lpf = LP_GROUP_NODES) ⊦ key-index(n0, n1) ∈ gc-heap(lpt, nx, n2 +1) ↔ n1 < n2 +1 ∧ ¬ n1 ∈ nx ∧ m = n0 ∧ lpf = LP_GROUP_NODES

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
lpt[n1] = mklp(m, m0, lpf, m1), n2 < # lpt, ∀ lpt, n1, m, m0, lpf, m1, n0, nx. lpt[n1] = mklp(m, m0, lpf, m1) ∧ n2 ≤ # lpt → (key-index(n0, n1) ∈ gc-heap(lpt, nx, n2) ↔ n1 < n2 ∧ ¬ n1 ∈ nx ∧ m = n0 ∧ lpf = LP_GROUP_NODES) ⊦ key-index(n0, n1) ∈ gc-heap(lpt, nx, n2) ∪ (¬ n2 ∈ nx ∧ lpt[n2].flags = LP_GROUP_NODES ⊃ ∅ ++ key-index(lpt[n2].ref-size, n2);∅) ↔ ¬ n2 < n1 ∧ ¬ n1 ∈ nx ∧ m = n0 ∧ lpf = LP_GROUP_NODES

Rule argument

Additional information

Used simplifier rules: gc-heap-rec: ⊦ 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)} ;∅ );% ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ {ki} = ∅ ++ ki; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m +1 = m + 1; (s)