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Proof step 18 from proof map-putex (previous step)

P(H0), r ∈ (∅ ++ r)[r, o], (∅ ++ r)[r, o] ⊥ H0 ⊦ r = null, ((r => o1 * P))((∅ ++ r)[r, o][r, o1] ∪ H0), r ∈ H0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
P(H0), r ∈ ∅ ++ r ⊦ ((r => o1 * P))((∅ ++ r)[r, o1] ∪ H0), r ∈ H0

Rule argument

Additional information

Used simplifier rules: ⊦ H[r, o0][r, o1] = H[r, o1]; (s)% ⊦ r ∈ H[r0, o] ↔ r ∈ H; (s)%r ≠ null ⊦ (∅ ++ r)[r, o] ⊥ H ↔ ¬ r ∈ H; (s)% ⊦ r ∈ H → r ≠ null; (s)