Nuprl Lemma : eq_mset_iff_eq_counts
∀s:DSet. ∀a,b:MSet{s}. (a = b ∈ MSet{s} ⇐⇒ ∀x:|s|. ((x #∈ a) = (x #∈ b) ∈ ℤ))
Proof
Definitions occuring in Statement :
mset_count: x #∈ a,
mset: MSet{s},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ,
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
dset: DSet,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
implies: P ⇒ Q,
sq_stable: SqStable(P),
mset: MSet{s},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
prop: ℙ,
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
squash: ↓T,
mk_mset: mk_mset(as),
mset_count: x #∈ a
Lemmas referenced :
sq_stable__iff,
equal_wf,
mset_wf,
all_wf,
set_car_wf,
mset_count_wf,
nat_wf,
sq_stable__equal,
sq_stable__all,
squash_wf,
iff_wf,
list_wf,
subtype_quotient,
permr_wf,
permr_equiv_rel,
equal-wf-base,
dset_wf,
equal_mset_elim,
mk_mset_wf,
permr_iff_eq_counts
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
lambdaEquality,
intEquality,
applyEquality,
because_Cache,
independent_functionElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
imageMemberEquality,
baseClosed,
productEquality,
addLevel,
independent_pairFormation,
impliesFunctionality
Latex:
\mforall{}s:DSet. \mforall{}a,b:MSet\{s\}. (a = b \mLeftarrow{}{}\mRightarrow{} \mforall{}x:|s|. ((x \#\mmember{} a) = (x \#\mmember{} b)))
Date html generated:
2017_10_01-AM-09_59_48
Last ObjectModification:
2017_03_03-PM-01_01_11
Theory : mset
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