Nuprl Lemma : bag-count-member
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[b:bag(T)]. uiff(1 ≤ (#x in b);x ↓∈ b)
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
deq: EqDecider(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
bag-member: x ↓∈ bs,
nat: ℕ,
le: A ≤ B,
not: ¬A,
false: False,
bag-filter: [x∈b|p[x]],
bag-size: #(bs),
ge: i ≥ j ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
true: True,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
bag-count-sqequal,
bag_to_squash_list,
sq_stable__bag-member,
le_wf,
bag-size_wf,
assert_wf,
bag-filter_wf,
bag-member_wf,
nat_wf,
sq_stable__le,
less_than'_wf,
bag-count_wf,
bag_wf,
deq_wf,
length_of_not_nil,
filter_wf5,
l_member_wf,
member_exists,
member_filter,
bag-member-list,
decidable-equal-deq,
and_wf,
equal_wf,
iff_wf,
squash_wf,
sq_stable__all,
sq_stable__iff,
sq_stable__equal,
sq_stable__assert,
assert_witness,
l_member_decomp,
length_wf_nat,
filter_append,
append_wf,
cons_wf,
nil_wf,
true_wf,
length_append,
subtype_rel_list,
top_wf,
iff_weakening_equal,
length_wf,
filter_cons_lemma,
filter_nil_lemma,
bool_wf,
equal-wf-T-base,
length_of_cons_lemma,
length_of_nil_lemma,
non_neg_length,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
bnot_wf,
not_wf,
eqof_wf,
le_weakening2,
add_functionality_wrt_eq,
uiff_transitivity,
eqtt_to_assert,
safe-assert-deq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalTransitivity,
computationStep,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
independent_pairFormation,
isect_memberFormation,
imageElimination,
independent_functionElimination,
productElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
natural_numberEquality,
setEquality,
cumulativity,
applyEquality,
setElimination,
rename,
lambdaEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
dependent_functionElimination,
voidElimination,
axiomEquality,
equalityTransitivity,
universeEquality,
isect_memberEquality,
independent_isectElimination,
lambdaFormation,
dependent_set_memberEquality,
functionExtensionality,
intEquality,
voidEquality,
addEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
equalityElimination,
addLevel,
impliesFunctionality,
levelHypothesis
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[b:bag(T)]. uiff(1 \mleq{} (\#x in b);x \mdownarrow{}\mmember{} b)
Date html generated:
2018_05_21-PM-09_52_02
Last ObjectModification:
2017_07_26-PM-06_31_35
Theory : bags_2
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