Nuprl Lemma : bag-parts'-no-repeats
∀[T:Type]
∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)]. bag-no-repeats(bag(T) List+;bag-parts'(eq;bs;x)) supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-parts': bag-parts'(eq;bs;x),
bag-no-repeats: bag-no-repeats(T;bs),
bag: bag(T),
listp: A List+,
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
bag-parts': bag-parts'(eq;bs;x),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
listp: A List+,
so_lambda: λ2x.t[x],
so_apply: x[s],
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
subtype_rel: A ⊆r B,
nat: ℕ,
cand: A c∧ B,
inject: Inj(A;B;f),
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
cons: [a / b],
top: Top,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
bag_wf,
deq_wf,
valueall-type_wf,
bag-null_wf,
bool_wf,
eqtt_to_assert,
assert-bag-null,
bag-single-no-repeats,
listp_wf,
cons_wf_listp,
nil_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
valueall-type-has-valueall,
bag-valueall-type,
set-valueall-type,
list_wf,
less_than_wf,
length_wf,
list-valueall-type,
bag-parts_wf,
evalall-reduce,
bag-append-no-repeats,
bag-map_wf,
empty-bag_wf,
bag-filter_wf,
eq_int_wf,
bag-count_wf,
hd_wf,
listp_properties,
nat_wf,
subtype_rel_bag,
assert_wf,
bag-member_wf,
bag-map-no-repeats,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
reduce_tl_cons_lemma,
and_wf,
tl_wf,
bag-parts-no-repeats,
bag-filter-no-repeats,
bag-member-map,
bag-member-filter,
bag-member-parts,
l_all_iff,
not_wf,
l_member_wf,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
cumulativity,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
voidElimination,
lambdaEquality,
natural_numberEquality,
callbyvalueReduce,
setElimination,
rename,
applyEquality,
setEquality,
independent_pairFormation,
hypothesis_subsumption,
voidEquality,
addEquality,
intEquality,
minusEquality,
dependent_set_memberEquality,
applyLambdaEquality,
inlFormation,
hyp_replacement
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[bs:bag(T)]. bag-no-repeats(bag(T) List\msupplus{};bag-parts'(eq;bs;x))
supposing valueall-type(T)
Date html generated:
2018_05_21-PM-09_51_34
Last ObjectModification:
2017_07_26-PM-06_31_32
Theory : bags_2
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