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Proof step 3 from proof mapex-new (previous step)

⊦ P(mk(rs, f)) ↔ ((r =>_ * P))(mk(rs ++ r, f)), r = null, r ∈ rs

User applied rule apply lemma

And obtained the following new goals

Proof step 4:
¬ r ∈ H, r ≠ null ⊦ P(H) ↔ ((r =>_ * P))(H ++ r)
Proof step 5:
P(mk(rs, f)) ↔ ((r =>_ * P))(mk(rs, f) ++ r) ⊦ P(mk(rs, f)) ↔ ((r =>_ * P))(mk(rs ++ r, f)), r = null, r ∈ rs

Rule argument

spec: maplet name: mapex-new seq: ¬ r ∈ H, r ≠ null ⊦ P(H) ↔ ((r =>_ * P))(H ++ r) subst: [r, H, P] → [r, mk(rs, f), P]

Additional information

Used simplifier rules: in-mk: ⊦ r ∈ mk(rs, f) ↔ r ∈ rs; Used terms: r, mk(rs, f), P