Nuprl Lemma : bag-settype
∀[T:Type]. ∀[bs:bag(T)]. ∀[P:T ⟶ ℙ]. bs ∈ bag({x:T| P[x]} ) supposing ∀x:T. (x ↓∈ bs ⇒ P[x])
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag: bag(T),
all: ∀x:A. B[x],
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
list_wf,
permutation_wf,
permutation_weakening,
list-set-type2,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
bag-member-select,
int_seg_wf,
equal-wf-base,
member_wf,
squash_wf,
true_wf,
bag_wf,
all_wf,
bag-member_wf,
list-subtype-bag,
subtype_rel_self,
permutation-strong-subtype,
strong-subtype-set2,
quotient-member-eq,
permutation-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
promote_hyp,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
independent_isectElimination,
pointwiseFunctionality,
sqequalRule,
pertypeElimination,
productElimination,
lambdaEquality,
applyEquality,
cumulativity,
setElimination,
rename,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
imageElimination,
productEquality,
setEquality,
imageMemberEquality,
baseClosed,
axiomEquality,
functionEquality,
universeEquality,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. bs \mmember{} bag(\{x:T| P[x]\} ) supposing \mforall{}x:T. (x \mdownarrow{}\mmember{} bs {}\mRightarrow{} P[x])
Date html generated:
2018_05_21-PM-06_25_07
Last ObjectModification:
2018_05_19-PM-05_15_36
Theory : bags
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