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Proof step 9 from proof at-uni (previous step)

r0 ∈ H1, H0 ⊥ H1, r ∈ H0 ∨ r ∈ H1, ∀ H, H0, r. # H < # H1 → H0 ⊥ H ∧ (r ∈ H0 ∨ r ∈ H) → (H0 ∪ H)[r] = (r ∈ H0 ⊃ H0[r];H[r]) ⊦ (H0 ∪ H1)[r] = (r ∈ H0 ⊃ H0[r];H1[r])

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 ⊥ H1, ∃ H0, o. H1 = (H0 ++ r0)[r0, o] ∧ ¬ r0 ∈ H0, r ∈ H0 ∨ r ∈ H1, ∀ H, H0, r. # H < # H1 → H0 ⊥ H ∧ (r ∈ H0 ∨ r ∈ H) → (H0 ∪ H)[r] = (r ∈ H0 ⊃ H0[r];H[r]) ⊦ (H0 ∪ H1)[r] = (r ∈ H0 ⊃ H0[r];H1[r]), r0 = null

Rule argument

spec: <toplevel> name: exin seq: ⊦ r ≠ null → (r ∈ H ↔ (∃ H0, o. H = (H0 ++ r)[r, o] ∧ ¬ r ∈ H0)) subst: [H, r] → [H1, r0] flags: <all paths>

Additional information

Used simplifier rules: sanity: ⊦ ¬ null ∈ H; Used terms: H1, r0