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Proof step 9 from proof prepost-gcheap-add-log (previous step)

keyset(gch0) = gc-heap(lpt1, nx), keyset(gch) = gc-heap(lpt1, log0), log0 = n0 + nx0, n0 < # lpt, bin-heap-inv(gch, lpt), bin-heap-inv(gch0, lpt1), id-idx(lpt1, lpt), ∀ n. n ∈ log0 → n < # lpt ∧ lpt[n].flags = LP_GROUP_NODES, ∀ n. n ∈ nx → n < # lpt1 ∧ lpt1[n].flags = LP_GROUP_NODES ⊦ ⟪gc_heap_insert#(key-index(lpt[n0].ref-size, n0); gch, lpt); log := nx0⟫ ( bin-heap-inv(gch, lpt) ∧ keyset(gch) = gc-heap(lpt1, log) ∧ id-idx(lpt1, lpt) ∧ ¬ dups(log) ∧ (∀ n. n ∈ log → n < # lpt ∧ lpt[n].flags = LP_GROUP_NODES) ∧ # log < # log0), log0 = [], dups(nx), dups(log0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
keyset(gch0) = gc-heap(lpt1, nx), keyset(gch) = gc-heap(lpt1, n0 ' + nx0), n0 < # lpt, bin-heap-inv(gch0, lpt1), bin-heap-inv(gch, lpt), id-idx(lpt1, lpt), ¬ dups(nx), ¬ dups(nx0), ¬ n0 ∈ nx0, ∀ n. n ∈ nx → n < # lpt1 ∧ lpt1[n].flags = LP_GROUP_NODES, ∀ n. n = n0 ∨ n ∈ nx0 → n < # lpt ∧ lpt[n].flags = LP_GROUP_NODES ⊦ ⟪gc_heap_insert#(key-index(lpt[n0].ref-size, n0); gch, lpt); log := nx0⟫ ( bin-heap-inv(gch, lpt) ∧ keyset(gch) = gc-heap(lpt1, log) ∧ id-idx(lpt1, lpt) ∧ ¬ dups(log) ∧ (∀ n. n ∈ log → n < # lpt ∧ lpt[n].flags = LP_GROUP_NODES) ∧ ¬ # nx0 < # log)

Rule argument

Additional information

Used simplifier rules: 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)% ⊦ #(n ' + nx) = (# nx)+1; (s)% ⊦ n ∈ n0 ' + nx ↔ n = n0 ∨ n ∈ nx; (s)% ⊦ dups(n ' + nx) ↔ dups(nx) ∨ n ∈ nx; (ws)%a: ⊦ (nx + nx0) + nx1 = nx + nx0 + nx1; from specification list[nat-list-set]% ⊦ n ' + nx ≠ []; (s)% ⊦ n + nx = n ' + nx; (s)