Nuprl Lemma : mset_mem_diff
∀s:DSet. ∀as:FiniteSet{s}. ∀bs:MSet{s}. ∀c:|s|. c ∈b as - bs = (c ∈b as) ∧b (¬b(c ∈b bs))
Proof
Definitions occuring in Statement :
mset_diff: a - b,
mset_mem: mset_mem,
finite_set: FiniteSet{s},
mset: MSet{s},
band: p ∧b q,
bnot: ¬bb,
bool: 𝔹,
all: ∀x:A. B[x],
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
dset: DSet,
mk_mset: mk_mset(as),
mset_diff: a - b,
mset_mem: mset_mem,
so_lambda: λ2x.t[x],
dislist: DisList{s},
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
finite_set: FiniteSet{s},
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff
Lemmas referenced :
set_car_wf,
list_wf,
dislist_wf,
all_mset_elim,
all_wf,
equal_wf,
bool_wf,
mset_mem_wf,
mset_diff_wf,
mk_mset_wf,
band_wf,
bnot_wf,
mset_wf,
sq_stable__all,
sq_stable__equal,
all_fset_elim,
finite_set_wf,
mem_wf,
diff_wf,
eqtt_to_assert,
dset_wf,
mem_diff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
dependent_functionElimination,
addLevel,
sqequalRule,
allFunctionality,
lambdaEquality,
because_Cache,
independent_functionElimination,
productElimination,
levelHypothesis,
allLevelFunctionality,
unionElimination,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}s:DSet. \mforall{}as:FiniteSet\{s\}. \mforall{}bs:MSet\{s\}. \mforall{}c:|s|. c \mmember{}\msubb{} as - bs = (c \mmember{}\msubb{} as) \mwedge{}\msubb{} (\mneg{}\msubb{}(c \mmember{}\msubb{} bs))
Date html generated:
2017_10_01-AM-10_00_09
Last ObjectModification:
2017_03_03-PM-01_01_42
Theory : mset
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