Nuprl Lemma : member-eclass-eclass2-eclass3-eclass1
∀[Info,A,B,C,D:Type]. ∀[X:EClass(A)]. ∀[Y:EClass(B)]. ∀[Z:EClass(C)]. ∀[f:Id ⟶ A ⟶ B ⟶ C ⟶ bag(D)]. ∀[es:EO+(Info)].
∀[e:E].
(uiff(↑e ∈b (eclass3((f o X);Y) o Z);(↑e ∈b X)
∧ (↑e ∈b Y)
∧ (↑e ∈b Z)
∧ (¬↑bag-null(f loc(e) X@e Y@e Z@e)))) supposing
(single-valued-classrel(es;X;A) and
single-valued-classrel(es;Y;B) and
single-valued-classrel(es;Z;C))
Proof
Definitions occuring in Statement :
eclass3: eclass3(X;Y),
eclass2: (X o Y),
eclass1: (f o X),
classfun-res: X@e,
single-valued-classrel: single-valued-classrel(es;X;T),
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,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
bag-null: bag-null(bs),
bag: bag(T)
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
cand: A c∧ B,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s]
Latex:
\mforall{}[Info,A,B,C,D:Type]. \mforall{}[X:EClass(A)]. \mforall{}[Y:EClass(B)]. \mforall{}[Z:EClass(C)]. \mforall{}[f:Id
{}\mrightarrow{} A
{}\mrightarrow{} B
{}\mrightarrow{} C
{}\mrightarrow{} bag(D)].
\mforall{}[es:EO+(Info)]. \mforall{}[e:E].
(uiff(\muparrow{}e \mmember{}\msubb{} (eclass3((f o X);Y) o Z);(\muparrow{}e \mmember{}\msubb{} X)
\mwedge{} (\muparrow{}e \mmember{}\msubb{} Y)
\mwedge{} (\muparrow{}e \mmember{}\msubb{} Z)
\mwedge{} (\mneg{}\muparrow{}bag-null(f loc(e) X@e Y@e Z@e)))) supposing
(single-valued-classrel(es;X;A) and
single-valued-classrel(es;Y;B) and
single-valued-classrel(es;Z;C))
Date html generated:
2016_05_17-AM-11_15_24
Last ObjectModification:
2016_01_18-AM-00_13_51
Theory : process-model
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