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Proof step 63 from proof embed-split (previous step)

mk(rs, f, g) = mk(rs0, f0, g0) ∪ mk(rs1, f1, g1), rs0 ∩ rs1 = ∅, P0(mk(rs0, f0)), P1(mk(rs1, f1)), ∀ r. r ∈ rs0 → g0(r) = a, ∀ r. r ∈ rs1 → g1(r) = a ⊦ ∀ r. (r ∈ mk(rs, f) ↔ r ∈ mk(rs0, f0) ∪ mk(rs1, f1)) ∧ (r ∈ mk(rs, f) → mk(rs, f)[r] = (mk(rs0, f0) ∪ mk(rs1, f1))[r])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 64:
mk(rs, f, g) = mk(rs0, f0, g0) ∪ mk(rs1, f1, g1), rs0 ∩ rs1 = ∅, P0(mk(rs0, f0)), P1(mk(rs1, f1)), ∀ r. r ∈ rs1 → g1(r) = a, ∀ r. r ∈ rs0 → g0(r) = a ⊦ (r ∈ rs ↔ r ∈ mk(rs0, f0) ∪ mk(rs1, f1)) ∧ (r ∈ rs → f(r) = (mk(rs0, f0) ∪ mk(rs1, f1))[r])

Rule argument

Additional information

Used simplifier rules: ⊦ r ∈ rs → mk(rs, f)[r] = f(r); (s)% ⊦ r ∈ mk(rs, f) ↔ r ∈ rs; (s)