Nuprl Lemma : bag-combine-size-bound
∀[A,B:Type]. ∀[f:A ⟶ bag(B)]. ∀[L:A List]. ∀[a:A]. #(f[a]) ≤ #(⋃a∈L.f[a]) supposing (a ∈ L)
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
bag-size: #(bs),
bag: bag(T),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
le: A ≤ B,
bag-size: #(bs),
bag-sum: bag-sum(ba;x.f[x]),
less_than': less_than'(a;b),
not: ¬A,
false: False,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
or: P ∨ Q,
ge: i ≥ j ,
decidable: Dec(P),
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
cons: [a / b],
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b
Lemmas referenced :
list-subtype-bag,
le_wf,
squash_wf,
true_wf,
istype-int,
bag-size_wf,
bag-combine-size,
istype-nat,
subtype_rel_self,
iff_weakening_equal,
le_witness_for_triv,
l_member_wf,
list_wf,
bag_wf,
istype-universe,
istype-void,
istype-le,
list_induction,
nat_wf,
list_accum_wf,
list_accum_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
list_accum_cons_lemma,
cons_wf,
cons_member,
add_nat_wf,
nat_properties,
decidable__le,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_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_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-swap,
add-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesisEquality,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
universeIsType,
hypothesis,
sqequalRule,
imageElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
productElimination,
independent_functionElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
lambdaFormation_alt,
voidElimination,
equalityIstype,
dependent_functionElimination,
functionEquality,
intEquality,
addEquality,
Error :memTop,
unionElimination,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
hyp_replacement,
intWeakElimination,
functionIsTypeImplies,
hypothesis_subsumption,
sqequalBase
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} bag(B)]. \mforall{}[L:A List]. \mforall{}[a:A]. \#(f[a]) \mleq{} \#(\mcup{}a\mmember{}L.f[a]) supposing (a \mmember{} L)
Date html generated:
2020_05_20-AM-08_01_40
Last ObjectModification:
2019_12_31-PM-06_30_47
Theory : bags
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