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Proof step 55 from proof embed-split (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 56:
⊦ a[P](mk(rs, f, g)) ↔ P(mk(rs, f)) ∧ (∀ r. r ∈ rs → g(r) = a)
Proof step 57:
mk(rs, f, g) = mk(rs0, f0, g0) ∪ mk(rs1, f1, g1), rs0 ∩ rs1 = ∅, P0(mk(rs0, f0)), a[P1](mk(rs1, f1, g1)), ∀ r. r ∈ rs0 → g0(r) = a ⊦ mk(rs, f) = mk(rs0, f0) ∪ mk(rs1, f1) ∧ P0(mk(rs0, f0)) ∧ P1(mk(rs1, f1))

Rule argument

spec: <toplevel> name: lift-aux seq: ⊦ a[P](mk(rs, f, g)) ↔ P(mk(rs, f)) ∧ (∀ r. r ∈ rs → g(r) = a) subst: [g, f, rs, P, a] → [g0, f0, rs0, P0, a] flags: <all paths>

Additional information

Used terms: g0, f0, rs0, P0, a