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Proof step 9 from proof mapex-new (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ ((r =>_ * P))(H) ↔ ((r => o1 * P))(H[r, o1]) ∧ r ∈ H
Proof step 11:
⊦ ((r => f(r) * P))((mk(rs, f) ++ r)[r, f(r)]) ∧ r ∈ mk(rs, f) ++ r ↔ ((r => f(r) * P))(mk(rs ++ r, f)[r, f(r)]), r = null, r ∈ rs

Rule argument

spec: maplet name: map-put seq: ⊦ ((r =>_ * P))(H) ↔ ((r => o1 * P))(H[r, o1]) ∧ r ∈ H subst: [o1, H, P, r] → [f(r), mk(rs, f) ++ r, P, r] flags: <all paths>

Additional information

Used terms: f(r), mk(rs, f) ++ r, P, r