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Proof step 9 from proof mk-exists (previous step)

r ∈ H, ∀ H0. # H0 < # H → true → (∃ rs, f. H0 = mk(rs, f)) ⊦ ∃ rs, f. H = mk(rs, f)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ r ≠ null → (r ∈ H ↔ (∃ H0, o. H = (H0 ++ r)[r, o] ∧ ¬ r ∈ H0))
Proof step 11:
∃ H0, o. H = (H0 ++ r)[r, o] ∧ ¬ r ∈ H0, ∀ H0. # H0 < # H → true → (∃ rs, f. H0 = mk(rs, f)) ⊦ r = null, ∃ rs, f. H = mk(rs, f)

Rule argument

spec: heap name: exin seq: ⊦ r ≠ null → (r ∈ H ↔ (∃ H0, o. H = (H0 ++ r)[r, o] ∧ ¬ r ∈ H0)) subst: [H, r] → [H, r] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ¬ null ∈ H; (s) Used terms: H, r