Nuprl Lemma : member-bag-remove-repeats
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. ∀[x:T]. uiff(x ↓∈ bs;x ↓∈ bag-remove-repeats(eq;bs))
Proof
Definitions occuring in Statement :
bag-remove-repeats: bag-remove-repeats(eq;bs),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
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,
subtype_rel: A ⊆r B,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
true: True,
bag-member: x ↓∈ bs,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bag-member-count,
lt_int_wf,
bag-count_wf,
nat_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
false_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
less_than'_wf,
ifthenelse_wf,
le_wf,
decidable__le,
bag-member_wf,
bag-remove-repeats_wf,
uiff_wf,
bag_wf,
deq_wf,
iff_weakening_uiff,
count-bag-remove-repeats,
bool_cases,
iff_transitivity,
assert_of_bnot
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_pairFormation,
isect_memberFormation,
natural_numberEquality,
cumulativity,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
independent_pairEquality,
axiomEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
addLevel,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)]. \mforall{}[x:T]. uiff(x \mdownarrow{}\mmember{} bs;x \mdownarrow{}\mmember{} bag-remove-repeats(eq;bs))
Date html generated:
2018_05_21-PM-09_47_32
Last ObjectModification:
2017_07_26-PM-06_30_15
Theory : bags_2
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