Nuprl Lemma : mem_bsubmset
∀s:DSet. ∀a:FiniteSet{s}. ∀b:MSet{s}. (↑(a ⊆b b) ⇐⇒ ∀x:|s|. ((↑(x ∈b a)) ⇒ (↑(x ∈b b))))
Proof
Definitions occuring in Statement :
bsubmset: a ⊆b b,
mset_mem: mset_mem,
finite_set: FiniteSet{s},
mset: MSet{s},
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
finite_set: FiniteSet{s},
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
gt: i > j,
dset: DSet,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
ge: i ≥ j
Lemmas referenced :
int_formula_prop_eq_lemma,
intformeq_wf,
nat_properties,
decidable__le,
decidable__equal_int,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
all_functionality_wrt_iff,
fset_properties,
dset_wf,
finite_set_wf,
mset_wf,
le_wf,
set_car_wf,
all_wf,
iff_wf,
nat_wf,
mset_count_wf,
gt_wf,
bsubmset_wf,
mset_mem_wf,
assert_wf,
mset_mem_iff_count_nzero,
count_bsubmset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
addLevel,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
impliesFunctionality,
lemma_by_obid,
dependent_functionElimination,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_functionElimination,
allFunctionality,
because_Cache,
isectElimination,
applyEquality,
lambdaEquality,
sqequalRule,
natural_numberEquality,
allLevelFunctionality,
impliesLevelFunctionality,
functionEquality,
independent_isectElimination,
unionElimination,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
setEquality
Latex:
\mforall{}s:DSet. \mforall{}a:FiniteSet\{s\}. \mforall{}b:MSet\{s\}. (\muparrow{}(a \msubseteq{}\msubb{} b) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} a)) {}\mRightarrow{} (\muparrow{}(x \mmember{}\msubb{} b))))
Date html generated:
2016_05_16-AM-07_51_01
Last ObjectModification:
2016_01_16-PM-11_40_02
Theory : mset
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