Nuprl Lemma : sub-bag-no-repeats-iff
∀[T:Type]. ∀eq:EqDecider(T). ∀b1,b2:bag(T). (bag-no-repeats(T;b1) ⇒ (∀x:T. (x ↓∈ b1 ⇒ x ↓∈ b2) ⇐⇒ sub-bag(T;b1;b2)))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
sub-bag: sub-bag(T;as;bs),
bag-no-repeats: bag-no-repeats(T;bs),
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs,
squash: ↓T,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
guard: {T},
nat: ℕ,
ge: i ≥ j ,
sq_type: SQType(T),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
le: A ≤ B,
sub-bag: sub-bag(T;as;bs),
sq_or: a ↓∨ b
Lemmas referenced :
all_wf,
bag-member_wf,
sub-bag_wf,
bag-no-repeats_wf,
bag_wf,
deq_wf,
sub-bag-iff,
decidable__equal_int,
bag-count_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
le_wf,
decidable__le,
nat_properties,
set_wf,
intformle_wf,
int_formula_prop_le_lemma,
equal-wf-T-base,
subtype_base_sq,
int_subtype_base,
bag-no-repeats-count,
bag-count-member,
less_than'_wf,
bag-member-append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
applyEquality,
because_Cache,
natural_numberEquality,
unionElimination,
inlFormation,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
inrFormation,
setElimination,
rename,
instantiate,
independent_pairEquality,
axiomEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}b1,b2:bag(T).
(bag-no-repeats(T;b1) {}\mRightarrow{} (\mforall{}x:T. (x \mdownarrow{}\mmember{} b1 {}\mRightarrow{} x \mdownarrow{}\mmember{} b2) \mLeftarrow{}{}\mRightarrow{} sub-bag(T;b1;b2)))
Date html generated:
2016_10_25-AM-11_30_57
Last ObjectModification:
2016_07_12-AM-07_35_47
Theory : bags_2
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