[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 prefix-in (previous step)

st0[m] = mkblock(x, lax, m2), st1[m] = mkblock(x0, lax0, m3), @(m0, m1) ∈ lax, m ∈ st0, dom st0 ⊆ dom st1, m ∈ st1 → x ⊑ x0 ∧ lax ⊑ lax0 ∧ # x = # lax ∧ # x0 = # lax0, ∀ n. n ∈ st0 ∧ n ∈ st1 → st0[n].list ⊑ st1[n].list ∧ st0[n].ladrlist ⊑ st1[n].ladrlist ∧ # st0[n].list = # st0[n].ladrlist ∧ # st1[n].list = # st1[n].ladrlist ⊦ @(m, m0, m1) ∈ st1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ ns ⊆ ns0 ↔ (∀ n. n ∈ ns → n ∈ ns0)
Proof step 11:
st1[m] = mkblock(x0, lax0, m3), st0[m] = mkblock(x, lax, m2), m ∈ st0, @(m0, m1) ∈ lax, m ∈ st1 → x ⊑ x0 ∧ lax ⊑ lax0 ∧ # x = # lax ∧ # x0 = # lax0, ∀ n. n ∈ st0 ∧ n ∈ st1 → st0[n].list ⊑ st1[n].list ∧ st0[n].ladrlist ⊑ st1[n].ladrlist ∧ # st0[n].list = # st0[n].ladrlist ∧ # st1[n].list = # st1[n].ladrlist, ∀ n. n ∈ dom st0 → n ∈ dom st1 ⊦ @(m, m0, m1) ∈ st1

Rule argument

spec: set-basic, inst: nat-set name: Subset seq: ⊦ ns ⊆ ns0 ↔ (∀ n. n ∈ ns → n ∈ ns0) subst: [ns0, ns] → [dom st1, dom st0] flags: <all paths>

Additional information

Used terms: dom st1, dom st0