Nuprl Lemma : sub-bag-filter2
āT:Type. āp1,p2:T ā¶ š¹. āb,c:bag(T).
(sub-bag(T;b;[xāc|p1[x]])
ā sub-bag(T;b;[xāc|p2[x]])
ā sub-bag(T;b;[xāc|p1[x] ā§b p2[x]]))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs)
,
bag-filter: [xāb|p[x]]
,
bag: bag(T)
,
band: p ā§b q
,
bool: š¹
,
so_apply: x[s]
,
all: āx:A. B[x]
,
implies: P
ā Q
,
function: x:A ā¶ B[x]
,
universe: Type
Definitions unfolded in proof :
all: āx:A. B[x]
,
implies: P
ā Q
,
uall: ā[x:A]. B[x]
,
member: t ā T
,
so_lambda: Ī»2x.t[x]
,
so_apply: x[s]
,
iff: P
āā Q
,
and: P ā§ Q
,
rev_implies: P
ā Q
,
cand: A cā§ B
,
uiff: uiff(P;Q)
,
rev_uimplies: rev_uimplies(P;Q)
,
uimplies: b supposing a
,
prop: ā
,
subtype_rel: A ār B
,
guard: {T}
Lemmas referenced :
sub-bag-filter,
band_wf,
assert_of_band,
bag-member_wf,
sub-bag_wf,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
bag_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
productElimination,
independent_functionElimination,
independent_pairFormation,
independent_isectElimination,
setEquality,
setElimination,
rename,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}T:Type. \mforall{}p1,p2:T {}\mrightarrow{} \mBbbB{}. \mforall{}b,c:bag(T).
(sub-bag(T;b;[x\mmember{}c|p1[x]]) {}\mRightarrow{} sub-bag(T;b;[x\mmember{}c|p2[x]]) {}\mRightarrow{} sub-bag(T;b;[x\mmember{}c|p1[x] \mwedge{}\msubb{} p2[x]]))
Date html generated:
2016_05_15-PM-02_45_22
Last ObjectModification:
2015_12_27-AM-09_37_27
Theory : bags
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