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
Lemmas :
es-interface-extensionality,
mapfilter-class_wf,
is-mapfilter-class,
es-E-interface-property,
es-interface-subtype_rel2,
es-E_wf,
event-ordering+_subtype,
assert_wf,
in-eclass_wf,
top_wf,
all_wf,
event-ordering+_wf,
iff_wf,
eclass-val_wf,
eclass_wf,
bool_wf,
mapfilter-class-val,
eqtt_to_assert,
bag_size_single_lemma,
false_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bag_size_empty_lemma
\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:
2015_07_17-PM-01_07_49
Last ObjectModification:
2015_01_27-PM-10_39_26
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