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Proof step 4 from proof del-lists-unions (previous step)

del-list(r ' + rx0, n)(H0), H0 ⊥ H1, ∀ n, H0, H1, m, P. del-list(rx0, n)(H0) ∧ H0 ⊥ H1 → (((del-list(rx0, m) * P))(H0 ∪ H1) ↔ n = m ∧ P(H1)) ⊦ ((del-list(r ' + rx0, m) * P))(H0 ∪ H1) ↔ n = m ∧ P(H1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ del-list(r ' + rx, n) = _∃(λ x. ls(r, x) * del-list(rx, n - # x) ∧ ⌜r ≠ null ∧ n ≥ # x⌝ )
Proof step 6:
(_∃(λ x. ls(r, x) * del-list(rx0, n - # x) ∧ ⌜r ≠ null ∧ n ≥ # x⌝ ))(H0), H0 ⊥ H1, ∀ n, H0, H1, m, P. del-list(rx0, n)(H0) ∧ H0 ⊥ H1 → (((del-list(rx0, m) * P))(H0 ∪ H1) ↔ n = m ∧ P(H1)) ⊦ ((del-list(r ' + rx0, m) * P))(H0 ∪ H1) ↔ n = m ∧ P(H1)

Rule argument

spec: <toplevel> name: del-list-rec seq: ⊦ del-list(r ' + rx, n) = _∃ (λ x. ls(r, x) * del-list(rx, n - # x) ∧ ⌜r ≠ null ∧ n ≥ # x⌝ ) subst: [n, rx, r] → [n, rx0, r] flags: <all paths>

Additional information

Used terms: n, rx0, r