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Proof step 17 from proof mk-exists (previous step)

∀ H1. # H1 < # (mk(rs, f) ++ r)[r, o] → true → (∃ rs, f. H1 = mk(rs, f)) ⊦ r = null, r ∈ rs, ∀ r0. (r0 ∈ (mk(rs, f) ++ r)[r, o] ↔ r0 ∈ mk(rs ++ r, f(r; o))) ∧ (r0 ∈ (mk(rs, f) ++ r)[r, o] → (mk(rs, f) ++ r)[r, o][r0] = mk(rs ++ r, f(r; o))[r0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: in-mk: ⊦ r ∈ mk(rs, f) ↔ r ∈ rs;% ⊦ H[r0, o][r] = (r = r0 ∧ r0 ∈ H ⊃ o;H[r]); (s)% ⊦ r ∈ rs ++ r0 ↔ r = r0 ∨ r ∈ rs; (s)%r0 ≠ null, ¬ r0 ∈ H ⊦ r ∈ H ++ r0 ↔ r = r0 ∨ r ∈ H; (s)% ⊦ r ∈ H[r0, o] ↔ r ∈ H; (s)