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Proof step 11 from proof del-lists-unions (previous step)

((ls(r, x) * del-list(rx0, n0)))(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)) ⊦ ((_∃(λ x. ls(r, x) * del-list(rx0, m - # x) ∧ ⌜r ≠ null ∧ m ≥ # x⌝ ) * P))(H0 ∪ H1) ↔ n0 + # x = m ∧ P(H1), r = null

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
((ls(r, x) * del-list(rx0, n0)))(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)) ⊦ (∃ x0. # x0 ≤ m ∧ (((ls(r, x0) * del-list(rx0, m - # x0)) * P))(H0 ∪ H1)) ↔ n0 + # x = m ∧ P(H1), r = null

Rule argument

Additional information

Used simplifier rules: ex-p: ⊦ ((_∃ xP * P))(H) ↔ (∃ x. ((xP(x) * P))(H));% ⊦ ⌜bv⌝ (H) ↔ bv; (s)% ⊦ ((P0 ∧ P1))(H) ↔ P0(H) ∧ P1(H); (s)%sep-com: ⊦ P0 * P1 = P1 * P0; from specification heap-sep%sep-ass: ⊦ (P0 * P1) * P2 = P0 * P1 * P2; from specification heap-sep%and-com: ⊦ (P0 ∧ P1) = (P1 ∧ P0); from specification heap-sep%and-ass: ⊦ ((P0 ∧ P1) ∧ P2) = (P0 ∧ P1 ∧ P2); from specification heap-sep% ⊦ (⌜bv⌝ ∧ P0) * P1 = (⌜bv⌝ ∧ P0 * P1); (s)% ⊦ n ≥ m ↔ m ≤ n; (s)