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Proof step 43 from proof precise-ls (previous step)

H4[r] = a × r1, ((r => a × r0))(H4), ls(r0, x0)(H3), ls(r1, x0)(H2), r ∈ ((∅ ++ r)[r, a × r1] ∪ H2) ∪ H, r ∈ (H4 ∪ H3) ∪ H0, H ⊥ H2, H4 ⊥ H3, H0 ⊥ H4, H0 ⊥ H3, ∀ H2, H0, H1, H, r. H2 ∪ H0 = H1 ∪ H ∧ ls(r, x0)(H2) ∧ ls(r, x0)(H1) ∧ H2 ⊥ H0 ∧ H1 ⊥ H → H2 = H1, ∀ r0. (r0 ∈ (H4 ∪ H3) ∪ H0 ↔ r0 ∈ ((∅ ++ r)[r, a × r1] ∪ H2) ∪ H) ∧ (r0 ∈ (H4 ∪ H3) ∪ H0 → ((H4 ∪ H3) ∪ H0)[r0] = (((∅ ++ r)[r, a × r1] ∪ H2) ∪ H)[r0]) ⊦ H4 ∪ H3 = (∅ ++ r)[r, a × r1] ∪ H2, r ∈ H2, r ∈ H, H4 = (∅ ++ r)[r, a × r1] ∧ r0 = r1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 44:
⊦ ((r => ce))(H) ↔ H = (∅ ++ r)[r, ce] ∧ r ≠ null

Rule argument

spec: maplet, inst: cell-heap name: maplet seq: ⊦ ((r => ce))(H) ↔ H = (∅ ++ r)[r, ce] ∧ r ≠ null subst: [H, ce, r] → [H4, a × r0, r] flags: <all paths>

Additional information

Used simplifier rules: ⊦ H[r0, ce][r] = (r = r0 ∧ r0 ∈ H ⊃ ce;H[r]); (s)% ⊦ ¬ r ∈ ∅; (s)%r0 ≠ null, ¬ r0 ∈ H ⊦ r ∈ H ++ r0 ↔ r = r0 ∨ r ∈ H; (s)% ⊦ a × r = a0 × r0 ↔ a = a0 ∧ r = r0; (ws)% ⊦ r ∈ H → r ≠ null; (s)%r ≠ null ⊦ (∅ ++ r)[r, ce] ⊥ H ↔ ¬ r ∈ H; (s)%r ≠ null, ¬ r ∈ H1 ⊦ H0 ⊥ (H1 ++ r)[r, ce] ↔ H0 ⊥ H1 ∧ ¬ r ∈ H0; (ws)% ⊦ H ⊥ ∅; (s) Used terms: H4, a × r0, r