Nuprl Lemma : bag-member-sv-bag-only
∀[T:Type]. ∀[b:bag(T)]. (sv-bag-only(b) ↓∈ b) supposing (0 < #(b) and single-valued-bag(b;T))
Proof
Definitions occuring in Statement :
sv-bag-only: sv-bag-only(b),
single-valued-bag: single-valued-bag(b;T),
bag-member: x ↓∈ bs,
bag-size: #(bs),
bag: bag(T),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag-member: x ↓∈ bs,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
exists: ∃x:A. B[x],
and: P ∧ Q,
sv-bag-only: sv-bag-only(b),
bag-size: #(bs),
single-valued-bag: single-valued-bag(b;T),
implies: P ⇒ Q,
bag: bag(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
ge: i ≥ j ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
cand: A c∧ B,
l_member: (x ∈ l)
Lemmas referenced :
less_than_wf,
bag-size_wf,
nat_wf,
single-valued-bag_wf,
bag_wf,
mklist_wf,
sv-bag-only_wf,
int_seg_wf,
equal_wf,
list-subtype-bag,
l_member_wf,
bag_to_squash_list,
quotient-member-eq,
list_wf,
permutation_wf,
permutation-equiv,
length_wf_nat,
hd_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
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,
all_wf,
bag-member_wf,
permutation_weakening,
list_extensionality,
mklist_length,
mklist_select,
squash_wf,
true_wf,
nat_properties,
lelt_wf,
select_wf,
iff_weakening_equal,
select0,
false_wf,
select_member,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
imageElimination,
hypothesis,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
natural_numberEquality,
cumulativity,
applyEquality,
lambdaEquality,
setElimination,
rename,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_pairFormation,
independent_isectElimination,
independent_pairFormation,
productEquality,
productElimination,
lambdaFormation,
dependent_functionElimination,
unionElimination,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
functionEquality,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[b:bag(T)]. (sv-bag-only(b) \mdownarrow{}\mmember{} b) supposing (0 < \#(b) and single-valued-bag(b;T))
Date html generated:
2017_10_01-AM-08_55_41
Last ObjectModification:
2017_07_26-PM-04_37_49
Theory : bags
Home
Index