Nuprl Lemma : mapfilter-class_functionality
∀[Info,A1,A2,B:Type]. ∀[P1:A1 ⟶ 𝔹]. ∀[P2:A2 ⟶ 𝔹]. ∀[f1:A1 ⟶ B]. ∀[f2:A2 ⟶ B]. ∀[X1:EClass(A1)]. ∀[X2:EClass(A2)].
(f1[v] where v from X1 such that P1[v]) = (f2[v] where v from X2 such that P2[v]) ∈ EClass(B)
supposing ∀es:EO+(Info). ∀e:E.
((↑e ∈b X1 ⇐⇒ ↑e ∈b X2)
∧ ((↑e ∈b X1)
⇒ (↑e ∈b X2)
⇒ ((↑P1[X1(e)] ⇐⇒ ↑P2[X2(e)]) ∧ ((↑P1[X1(e)]) ⇒ (↑P2[X2(e)]) ⇒ (f1[X1(e)] = f2[X2(e)] ∈ B)))))
Proof
Definitions occuring in Statement :
mapfilter-class: (f[v] where v from X such that P[v]),
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
es-E-interface: E(X),
prop: ℙ,
rev_implies: P ⇐ Q,
guard: {T},
sv-class: Singlevalued(X),
mapfilter-class: (f[v] where v from X such that P[v]),
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,
ifthenelse: if b then t else f fi ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Latex:
\mforall{}[Info,A1,A2,B:Type]. \mforall{}[P1:A1 {}\mrightarrow{} \mBbbB{}]. \mforall{}[P2:A2 {}\mrightarrow{} \mBbbB{}]. \mforall{}[f1:A1 {}\mrightarrow{} B]. \mforall{}[f2:A2 {}\mrightarrow{} B]. \mforall{}[X1:EClass(A1)].
\mforall{}[X2:EClass(A2)].
(f1[v] where v from X1 such that P1[v]) = (f2[v] where v from X2 such that P2[v])
supposing \mforall{}es:EO+(Info). \mforall{}e:E.
((\muparrow{}e \mmember{}\msubb{} X1 \mLeftarrow{}{}\mRightarrow{} \muparrow{}e \mmember{}\msubb{} X2)
\mwedge{} ((\muparrow{}e \mmember{}\msubb{} X1)
{}\mRightarrow{} (\muparrow{}e \mmember{}\msubb{} X2)
{}\mRightarrow{} ((\muparrow{}P1[X1(e)] \mLeftarrow{}{}\mRightarrow{} \muparrow{}P2[X2(e)])
\mwedge{} ((\muparrow{}P1[X1(e)]) {}\mRightarrow{} (\muparrow{}P2[X2(e)]) {}\mRightarrow{} (f1[X1(e)] = f2[X2(e)])))))
Date html generated:
2016_05_16-PM-10_29_46
Last ObjectModification:
2015_12_29-AM-11_10_37
Theory : event-ordering
Home
Index