Nuprl Lemma : filter-image_functionality
∀[Info,T,A,B:Type]. ∀[f:A ⟶ bag(T)]. ∀[g:B ⟶ bag(T)]. ∀[X:EClass(A)]. ∀[Y:EClass(B)].
f[X] = g[Y] ∈ EClass(T)
supposing ∀es:EO+(Info). ∀e:E. ((↑e ∈b X ⇐⇒ ↑e ∈b Y) ∧ ((↑e ∈b X) ⇒ (↑e ∈b Y) ⇒ ((f X(e)) = (g Y(e)) ∈ bag(T))))
Proof
Definitions occuring in Statement :
es-filter-image: f[X],
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
eclass: EClass(A[eo; e]),
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s],
es-filter-image: f[X],
eclass-compose1: f o X,
eclass-val: X(e),
in-eclass: e ∈b X,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
not: ¬A,
es-E-interface: E(X),
false: False
Latex:
\mforall{}[Info,T,A,B:Type]. \mforall{}[f:A {}\mrightarrow{} bag(T)]. \mforall{}[g:B {}\mrightarrow{} bag(T)]. \mforall{}[X:EClass(A)]. \mforall{}[Y:EClass(B)].
f[X] = g[Y]
supposing \mforall{}es:EO+(Info). \mforall{}e:E.
((\muparrow{}e \mmember{}\msubb{} X \mLeftarrow{}{}\mRightarrow{} \muparrow{}e \mmember{}\msubb{} Y) \mwedge{} ((\muparrow{}e \mmember{}\msubb{} X) {}\mRightarrow{} (\muparrow{}e \mmember{}\msubb{} Y) {}\mRightarrow{} ((f X(e)) = (g Y(e)))))
Date html generated:
2016_05_16-PM-10_31_19
Last ObjectModification:
2015_12_29-AM-11_05_57
Theory : event-ordering
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