Nuprl Lemma : bag-member-subtype-2
∀[A:Type]. ∀b:bag(A). ∀x:A. (x ↓∈ b ⇒ x ↓∈ b)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
bag-member: x ↓∈ bs,
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
l_all: (∀x∈L.P[x]),
cand: A c∧ B,
subtype_rel: A ⊆r B,
respects-equality: respects-equality(S;T),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
less_than: a < b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag: bag(T),
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
label: ...$L... t,
permutation: permutation(T;L1;L2)
Lemmas referenced :
bag-member_wf,
bag_wf,
istype-universe,
sq_stable__bag-member,
bag-subtype,
list-set-type2,
int_seg_wf,
length_wf,
select_member,
subtype-respects-equality,
list_wf,
list-subtype-bag,
l_member_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
l_member-settype,
respects-equality-bag,
respects-equality-set-trivial,
permutation_wf,
permutation_transitivity,
l_all_functionality_wrt_permutation,
permutation_inversion,
bag-member-select,
quotient-member-eq,
permutation-equiv,
inject_wf,
permute_list_wf,
strong-subtype-equal-lists,
strong-subtype-set3,
strong-subtype-self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
functionIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality,
setEquality,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
independent_isectElimination,
natural_numberEquality,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType,
equalityIstype,
because_Cache,
setElimination,
rename,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
setIsType,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
sqequalBase,
applyEquality
Latex:
\mforall{}[A:Type]. \mforall{}b:bag(A). \mforall{}x:A. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} x \mdownarrow{}\mmember{} b)
Date html generated:
2019_10_15-AM-11_02_10
Last ObjectModification:
2018_11_30-PM-00_12_51
Theory : bags
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