[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 9 from proof Sep-forall (previous step)

((adP0(a, d) * adP0 ⊗_ st0))(H), ∀ a0, H. a0 ∈ st0[a, d] ∧ adP0(a0, st0[a, d][a0])(H) → adP(a0, st0[a, d][a0])(H), ∀ st1, adP0, H, adP. # st1 < # st0[a, d] → ((adP0 ⊗_ st1))(H) ∧ (∀ a, H. a ∈ st1 ∧ adP0(a, st1[a])(H) → adP(a, st1[a])(H)) → ((adP ⊗_ st1))(H) ⊦ ((adP(a, d) * adP ⊗_ st0))(H), a ∈ st0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ ((P0 * P1))(H) ↔ (∃ H0, H1. H0 ⊥ H1 ∧ H = H0 ∪ H1 ∧ P0(H0) ∧ P1(H1))
Proof step 11:
∀ a0, H. a0 ∈ st0[a, d] ∧ adP0(a0, st0[a, d][a0])(H) → adP(a0, st0[a, d][a0])(H), ∃ H0, H1. H0 ⊥ H1 ∧ H = H0 ∪ H1 ∧ adP0(a, d)(H0) ∧ ((adP0 ⊗_ st0))(H1), ∀ st1, adP0, H, adP. # st1 < # st0[a, d] → ((adP0 ⊗_ st1))(H) ∧ (∀ a, H. a ∈ st1 ∧ adP0(a, st1[a])(H) → adP(a, st1[a])(H)) → ((adP ⊗_ st1))(H) ⊦ ((adP(a, d) * adP ⊗_ st0))(H), a ∈ st0

Rule argument

spec: heap-sep name: sep seq: ⊦ ((P0 * P1))(H) ↔ (∃ H0, H1. H0 ⊥ H1 ∧ H = H0 ∪ H1 ∧ P0(H0) ∧ P1(H1)) subst: [H, P1, P0] → [H, adP0 ⊗_ st0, adP0(a, d)] flags: <all paths>

Additional information

Used terms: H, adP0 ⊗_ st0, adP0(a, d)