Nuprl Lemma : assert-bag-has-no-repeats
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[b:bag(T)]. (↑bag-has-no-repeats(eq;b) ⇐⇒ bag-no-repeats(T;b))
Proof
Definitions occuring in Statement :
bag-has-no-repeats: bag-has-no-repeats(eq;b),
bag-no-repeats: bag-no-repeats(T;bs),
bag: bag(T),
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
bag-has-no-repeats: bag-has-no-repeats(eq;b),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
true: True,
guard: {T},
uiff: uiff(P;Q),
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
equal_wf,
bag-size_wf,
bag-remove-repeats_wf,
nat_wf,
squash_wf,
true_wf,
bag_wf,
bag-remove-repeats-if-no-repeats,
subtype_rel_self,
iff_weakening_equal,
bag-no-repeats_wf,
assert_of_eq_int,
assert_wf,
eq_int_wf,
iff_wf,
deq_wf,
bag-remove-repeats-no-repeats,
sub-bag_antisymmetry,
sub-bag-iff,
count-bag-remove-repeats,
lt_int_wf,
bag-count_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
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_less_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
nat_properties,
bag-size-partition,
int_subtype_base,
bag-size-as-summation,
bag-summation-equal-implies-all-equal-1,
bag-member_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
bag-count-member,
bag-member-remove-repeats
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
independent_pairFormation,
lambdaFormation,
introduction,
sqequalRule,
sqequalHypSubstitution,
imageElimination,
hypothesis,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
intEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_isectElimination,
natural_numberEquality,
instantiate,
productElimination,
independent_functionElimination,
addLevel,
universeEquality,
hyp_replacement,
applyLambdaEquality,
dependent_functionElimination,
unionElimination,
equalityElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
promote_hyp,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[b:bag(T)]. (\muparrow{}bag-has-no-repeats(eq;b) \mLeftarrow{}{}\mRightarrow{} bag-no-repeats(T;b))
Date html generated:
2019_10_16-AM-11_32_54
Last ObjectModification:
2018_08_25-AM-11_18_22
Theory : bags_2
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