Nuprl Lemma : bag-member-union
∀[T:Type]. ∀[x:T]. ∀[bbs:bag(bag(T))]. uiff(x ↓∈ bag-union(bbs);↓∃b:bag(T). (x ↓∈ b ∧ b ↓∈ bbs))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-union: bag-union(bbs),
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
uimplies: b supposing a,
bag-member: x ↓∈ bs,
all: ∀x:A. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
empty-bag: {},
concat: concat(ll),
bag-union: bag-union(bbs),
bag-append: as + bs,
sq_or: a ↓∨ b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
cons-bag: x.b,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
bag_to_squash_list,
bag_wf,
sq_stable__uiff,
bag-member_wf,
bag-union_wf,
squash_wf,
sq_stable__bag-member,
sq_stable__squash,
uiff_wf,
istype-universe,
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,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
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,
le_wf,
istype-nat,
exists_wf,
empty-bag_wf,
bag-member-empty,
reduce_nil_lemma,
reduce_cons_lemma,
sq_or_wf,
list-subtype-bag,
cons_wf,
bag-member-append,
bag-append_wf,
bag-member-cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
imageElimination,
sqequalRule,
productEquality,
independent_functionElimination,
productElimination,
promote_hyp,
rename,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
instantiate,
universeEquality,
lambdaFormation_alt,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
voidElimination,
independent_pairFormation,
functionIsTypeImplies,
unionElimination,
hypothesis_subsumption,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
equalityTransitivity,
baseApply,
closedConclusion,
applyEquality,
intEquality,
sqequalBase,
lambdaEquality,
cumulativity,
isect_memberFormation,
voidEquality,
isect_memberEquality,
productIsType,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[bbs:bag(bag(T))]. uiff(x \mdownarrow{}\mmember{} bag-union(bbs);\mdownarrow{}\mexists{}b:bag(T). (x \mdownarrow{}\mmember{} b \mwedge{} b \mdownarrow{}\mmember{} bbs))
Date html generated:
2019_10_15-AM-11_01_46
Last ObjectModification:
2019_06_25-PM-03_25_58
Theory : bags
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