Nuprl Lemma : bag-no-repeats-count
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. uiff(bag-no-repeats(T;bs);∀[x:T]. uiff(1 ≤ (#x in bs);(#x in bs) = 1 ∈ ℤ))
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-no-repeats: bag-no-repeats(T;bs),
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
deq: EqDecider(T),
bag-filter: [x∈b|p[x]],
bag-size: #(bs),
guard: {T},
istype: istype(T),
cand: A c∧ B,
bag-count: (#x in bs),
top: Top,
true: True
Lemmas referenced :
istype-le,
bag-count_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
le_witness_for_triv,
set_subtype_base,
le_wf,
int_subtype_base,
bag-no-repeats_wf,
bag_wf,
deq_wf,
istype-universe,
bag-count-sqequal,
bag-size_wf,
assert_wf,
bag-filter_wf,
equal-wf-base,
no-repeats-iff-count,
filter_functionality,
eta_conv,
bool_wf,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
length_wf,
length_wf_nat,
list_wf,
exists_wf,
squash_wf,
no_repeats_wf,
equal_wf,
list-subtype-bag,
equal-wf-T-base,
nat_wf,
uiff_wf,
uall_wf,
bag_to_squash_list,
less_than'_wf,
satisfiable-full-omega-tt,
set_wf,
subtype_rel_self,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
dependent_functionElimination,
because_Cache,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
universeIsType,
voidElimination,
productElimination,
equalityIstype,
intEquality,
baseClosed,
sqequalBase,
independent_pairEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
imageElimination,
imageMemberEquality,
isectIsType,
productIsType,
instantiate,
universeEquality,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
setEquality,
setIsType,
lambdaFormation_alt,
isect_memberFormation,
productEquality,
dependent_pairFormation,
lambdaEquality,
cumulativity,
computeAll,
voidEquality,
isect_memberEquality,
lambdaFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)].
uiff(bag-no-repeats(T;bs);\mforall{}[x:T]. uiff(1 \mleq{} (\#x in bs);(\#x in bs) = 1))
Date html generated:
2020_05_20-AM-09_04_13
Last ObjectModification:
2020_01_04-PM-10_27_42
Theory : bags_2
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