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Proof step 12 from proof mapex-new (previous step)

P(H) ⊦ r = null, r ∈ H, ∀ r0. (r0 ∈ H ++ r ↔ r0 ∈ (∅ ++ r)[r, (H ++ r)[r]] ∪ H) ∧ (r0 ∈ H ++ r → (H ++ r)[r0] = ((∅ ++ r)[r, (H ++ r)[r]] ∪ H)[r0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
P(H) ⊦ r = null, r ∈ H, (r0 = r ∨ r0 ∈ H ↔ r0 ∈ (∅ ++ r)[r, (H ++ r)[r]] ∪ H) ∧ (r0 = r ∨ r0 ∈ H → (H ++ r)[r0] = ((∅ ++ r)[r, (H ++ r)[r]] ∪ H)[r0])

Rule argument

Additional information

Used simplifier rules: r0 ≠ null, ¬ r0 ∈ H ⊦ r ∈ H ++ r0 ↔ r = r0 ∨ r ∈ H; (s)