[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 40 from proof precise-ls (previous step)

H4[r] = H5[r], ((r => a × r0))(H4), ((r => a × r1))(H5), ls(r0, x0)(H3), ls(r1, x0)(H2), r ∈ (H5 ∪ H2) ∪ H, r ∈ (H4 ∪ H3) ∪ H0, H4 ⊥ H3, H0 ⊥ H4, H0 ⊥ H3, H5 ⊥ H2, H ⊥ H5, H ⊥ H2, ∀ H2, H0, H1, H, r. H2 ∪ H0 = H1 ∪ H ∧ ls(r, x0)(H2) ∧ ls(r, x0)(H1) ∧ H2 ⊥ H0 ∧ H1 ⊥ H → H2 = H1, ∀ r. (r ∈ (H4 ∪ H3) ∪ H0 ↔ r ∈ (H5 ∪ H2) ∪ H) ∧ (r ∈ (H4 ∪ H3) ∪ H0 → ((H4 ∪ H3) ∪ H0)[r] = ((H5 ∪ H2) ∪ H)[r]) ⊦ H4 ∪ H3 = H5 ∪ H2, H4 = H5 ∧ r0 = r1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 41:
⊦ ((r => ce))(H) ↔ H = (∅ ++ r)[r, ce] ∧ r ≠ null
Proof step 42:
H5 = (∅ ++ r)[r, a × r1], H4[r] = H5[r], ((r => a × r0))(H4), ls(r1, x0)(H2), ls(r0, x0)(H3), r ∈ (H4 ∪ H3) ∪ H0, r ∈ (H5 ∪ H2) ∪ H, H ⊥ H2, H ⊥ H5, H5 ⊥ H2, H0 ⊥ H3, H0 ⊥ H4, H4 ⊥ H3, ∀ r. (r ∈ (H4 ∪ H3) ∪ H0 ↔ r ∈ (H5 ∪ H2) ∪ H) ∧ (r ∈ (H4 ∪ H3) ∪ H0 → ((H4 ∪ H3) ∪ H0)[r] = ((H5 ∪ H2) ∪ H)[r]), ∀ H2, H0, H1, H, r. H2 ∪ H0 = H1 ∪ H ∧ ls(r, x0)(H2) ∧ ls(r, x0)(H1) ∧ H2 ⊥ H0 ∧ H1 ⊥ H → H2 = H1 ⊦ r = null, H4 ∪ H3 = H5 ∪ H2, H4 = H5 ∧ r0 = r1

Rule argument

spec: maplet, inst: cell-heap-aux name: maplet seq: ⊦ ((r => ce))(H) ↔ H = (∅ ++ r)[r, ce] ∧ r ≠ null subst: [H, ce, r] → [H5, a × r1, r] flags: <all paths>

Additional information

Used terms: H5, a × r1, r