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)
Lemmas :
all_wf,
event-ordering+_wf,
es-E_wf,
event-ordering+_subtype,
iff_wf,
assert_wf,
in-eclass_wf,
es-interface-subtype_rel2,
top_wf,
bag_wf,
eclass-val_wf,
eclass_wf,
bool_wf,
eqtt_to_assert,
uiff_transitivity,
equal-wf-T-base,
bnot_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
es-E-interface-property
\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:
2015_07_17-PM-01_08_47
Last ObjectModification:
2015_01_27-PM-10_36_22
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