Nuprl Lemma : classfun-res-disjoint-union-comb-as-parallel-eclass1
∀[Info,A,B,S:Type]. ∀[es:EO+(Info)]. ∀[X1:EClass(A)]. ∀[X2:EClass(B)]. ∀[e:E]. ∀[f1:Id ─→ A ─→ S ─→ S]. ∀[f2:Id
─→ B
─→ S
─→ S].
∀[s:S].
((f1 + f2 loc(e) X1 (+) X2@e s) = ((f1 o X1) || (f2 o X2)@e s) ∈ S) supposing
(disjoint-classrel(es;A;X1;B;X2) and
single-valued-classrel(es;X1;A) and
single-valued-classrel(es;X2;B) and
((↑e ∈b X1) ∨ (↑e ∈b X2)))
Proof
Definitions occuring in Statement :
disjoint-union-tr: tr1 + tr2,
disjoint-union-comb: X (+) Y,
parallel-class: X || Y,
eclass1: (f o X),
classfun-res: X@e,
single-valued-classrel: single-valued-classrel(es;X;T),
disjoint-classrel: disjoint-classrel(es;A;X;B;Y),
member-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-loc: loc(e),
es-E: E,
Id: Id,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T
Lemmas :
classfun-res-parallel-class-left,
eclass1_wf,
member-eclass-eclass1,
event-ordering+_subtype,
eclass1-disjoint-classrel,
disjoint-classrel-symm,
disjoint-union-tr_wf,
es-loc_wf,
classfun-res-disjoint-union-comb-left,
classfun-res_wf,
eclass1-single-val,
iff_weakening_equal,
disj_un_tr_ap_inl_lemma,
classfun-res-eclass1,
classfun-res-parallel-class-right,
classfun-res-disjoint-union-comb-right,
disj_un_tr_ap_inr_lemma
Latex:
\mforall{}[Info,A,B,S:Type]. \mforall{}[es:EO+(Info)]. \mforall{}[X1:EClass(A)]. \mforall{}[X2:EClass(B)]. \mforall{}[e:E]. \mforall{}[f1:Id
{}\mrightarrow{} A
{}\mrightarrow{} S
{}\mrightarrow{} S].
\mforall{}[f2:Id {}\mrightarrow{} B {}\mrightarrow{} S {}\mrightarrow{} S]. \mforall{}[s:S].
((f1 + f2 loc(e) X1 (+) X2@e s) = ((f1 o X1) || (f2 o X2)@e s)) supposing
(disjoint-classrel(es;A;X1;B;X2) and
single-valued-classrel(es;X1;A) and
single-valued-classrel(es;X2;B) and
((\muparrow{}e \mmember{}\msubb{} X1) \mvee{} (\muparrow{}e \mmember{}\msubb{} X2)))
Date html generated:
2015_07_23-AM-11_29_26
Last ObjectModification:
2015_02_04-PM-04_44_36
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