Nuprl Lemma : bag-subtype2
∀[A:Type]. ∀P:A ⟶ ℙ. ∀b:bag({x:A| P[x]} ). ∀x:{x:A| P[x]} . (x ↓∈ b ⇐⇒ x ↓∈ b)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
bag-member: x ↓∈ bs,
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
bag: bag(T),
quotient: x,y:A//B[x; y],
cand: A c∧ B,
permutation: permutation(T;L1;L2),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
label: ...$L... t,
guard: {T}
Lemmas referenced :
bag-member_wf,
subtype_rel_bag,
set_wf,
bag_wf,
bag_to_squash_list,
sq_stable__bag-member,
member_wf,
list_wf,
subtype_rel_list,
permutation_wf,
permutation_inversion,
permute_list_wf,
list-eq-subtype2,
quotient-member-eq,
permutation-equiv,
l_member-settype,
equal_wf,
list-subtype-bag,
subtype_rel_self,
l_member_wf,
permutation-strong-subtype,
strong-subtype-set2,
bag-member-subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
setElimination,
rename,
hypothesis,
applyEquality,
setEquality,
functionExtensionality,
because_Cache,
sqequalRule,
independent_isectElimination,
lambdaEquality,
dependent_set_memberEquality,
universeEquality,
functionEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
independent_functionElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
pertypeElimination,
productEquality,
equalityTransitivity,
dependent_pairFormation
Latex:
\mforall{}[A:Type]. \mforall{}P:A {}\mrightarrow{} \mBbbP{}. \mforall{}b:bag(\{x:A| P[x]\} ). \mforall{}x:\{x:A| P[x]\} . (x \mdownarrow{}\mmember{} b \mLeftarrow{}{}\mRightarrow{} x \mdownarrow{}\mmember{} b)
Date html generated:
2017_10_01-AM-08_56_31
Last ObjectModification:
2017_07_26-PM-04_38_53
Theory : bags
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