Nuprl Lemma : bag-subtype
∀[A:Type]. ∀b:bag(A). (b ∈ bag({x:A| x ↓∈ b} ))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
bag: bag(T),
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
bag-member: x ↓∈ bs,
exists: ∃x:A. B[x],
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
bag_wf,
bag-member_wf,
list_wf,
permutation_wf,
equal_wf,
equal-wf-base,
list-subtype,
l_member_wf,
list-subtype-bag,
subtype_rel_list_set,
permutation-strong-subtype,
strong-subtype-set2,
quotient-member-eq,
permutation-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
sqequalRule,
hypothesis,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
rename,
dependent_functionElimination,
independent_functionElimination,
productEquality,
lambdaEquality,
axiomEquality,
universeEquality,
applyEquality,
independent_isectElimination,
dependent_pairFormation,
independent_pairFormation,
imageMemberEquality,
baseClosed,
setElimination
Latex:
\mforall{}[A:Type]. \mforall{}b:bag(A). (b \mmember{} bag(\{x:A| x \mdownarrow{}\mmember{} b\} ))
Date html generated:
2017_10_01-AM-08_56_26
Last ObjectModification:
2017_07_26-PM-04_38_48
Theory : bags
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