Nuprl Lemma : bag-member-decidable2
∀T:Type. ∀P:T ⟶ ℙ. ∀b:bag(T). ∀x:{x:T| P[x]} .
((∀x,y:{x:T| P[x]} . Dec(x = y ∈ {x:T| P[x]} )) ⇒ (∀x:{x:T| x ↓∈ b} . ∃y:{x:T| P[x]} . (x = y ∈ T)) ⇒ Dec(x ↓∈ b))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
decidable: Dec(P),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
exists: ∃x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
pi1: fst(t),
uimplies: b supposing a,
squash: ↓T,
istype: istype(T),
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
and: P ∧ Q,
or: P ∨ Q,
empty-bag: {},
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
cons-bag: x.b,
bag-map': bag-map'(f;b),
single-bag: {x},
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bag-map: bag-map(f;bs),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
sq_or: a ↓∨ b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-universe,
bag-member_wf,
decidable_wf,
equal_wf,
subtype_rel_self,
bag_wf,
bag-map'_wf,
subtype_rel_bag,
bag_to_squash_list,
bag-subtype,
bag_qinc,
list-subtype-bag,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
list-cases,
empty-bag_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
cons-bag_wf,
nat_wf,
bag_map_empty_lemma,
cons_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
bag-map-append,
single-bag_wf,
top_wf,
map_cons_lemma,
map_nil_lemma,
squash_wf,
true_wf,
bag-member-cons,
exists_wf,
subtype_rel_dep_function,
subtype_rel_sets,
subtype_rel-equal,
all_wf,
bag-subtype2,
decidable_functionality,
decidable__bag-member2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
functionIsType,
setIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
universeIsType,
productIsType,
inhabitedIsType,
equalityIsType1,
setElimination,
rename,
setEquality,
applyEquality,
instantiate,
universeEquality,
because_Cache,
dependent_pairFormation_alt,
lambdaEquality_alt,
functionExtensionality,
dependent_functionElimination,
dependent_set_memberEquality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_isectElimination,
imageElimination,
intWeakElimination,
natural_numberEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
axiomEquality,
functionIsTypeImplies,
unionElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality,
imageMemberEquality,
inlFormation_alt,
inrFormation_alt,
hyp_replacement
Latex:
\mforall{}T:Type. \mforall{}P:T {}\mrightarrow{} \mBbbP{}. \mforall{}b:bag(T). \mforall{}x:\{x:T| P[x]\} .
((\mforall{}x,y:\{x:T| P[x]\} . Dec(x = y)) {}\mRightarrow{} (\mforall{}x:\{x:T| x \mdownarrow{}\mmember{} b\} . \mexists{}y:\{x:T| P[x]\} . (x = y)) {}\mRightarrow{} Dec(x \mdownarrow{}\mmember{} b))
Date html generated:
2019_10_16-AM-11_30_06
Last ObjectModification:
2018_10_10-PM-07_22_58
Theory : bags_2
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