Nuprl Lemma : bag-no-repeats-le-bag-size
∀[T:Type]. ∀[locs,b:bag(T)]. #(b) ≤ #(locs) supposing bag-no-repeats(T;b) ∧ (∀x:T. (x ↓∈ b ⇒ x ↓∈ locs))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag-size: #(bs),
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
bag-size: #(bs),
nat: ℕ,
le: A ≤ B,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
or: P ∨ Q,
decidable: Dec(P),
cons: [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],
uiff: uiff(P;Q),
cons-bag: x.b,
rev_uimplies: rev_uimplies(P;Q),
sq_or: a ↓∨ b,
bag-member: x ↓∈ bs,
sq_stable: SqStable(P),
true: True,
iff: P ⇐⇒ Q,
l_member: (x ∈ l),
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
bag-append: as + bs,
rev_implies: P ⇐ Q,
empty-bag: {}
Lemmas referenced :
bag-member_wf,
bag_to_squash_list,
list-subtype-bag,
squash_wf,
le_wf,
bag-size_wf,
le_witness_for_triv,
bag-no-repeats_wf,
bag_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,
length_of_nil_lemma,
non_neg_length,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
nil_wf,
list_wf,
no_repeats_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
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,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
length_of_cons_lemma,
no_repeats_cons,
bag-member-cons,
sq_stable__le,
length_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
cons_wf,
istype-nat,
list_decomp_member,
equal_wf,
append_wf,
not-list-member-not-bag-member,
bag-member-append,
bag-append_wf,
sq_stable__sq_or,
bag-member-empty-iff,
length_wf_nat,
length_append,
subtype_rel_list,
top_wf,
add-is-int-iff,
false_wf,
add_functionality_wrt_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
imageElimination,
equalitySymmetry,
hypothesis,
hyp_replacement,
applyLambdaEquality,
sqequalRule,
functionEquality,
hypothesisEquality,
extract_by_obid,
isectElimination,
promote_hyp,
applyEquality,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
inhabitedIsType,
rename,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
universeIsType,
setElimination,
equalityTransitivity,
productIsType,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
universeEquality,
lambdaFormation_alt,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
functionIsTypeImplies,
unionElimination,
voidEquality,
closedConclusion,
hypothesis_subsumption,
equalityIstype,
dependent_set_memberEquality_alt,
baseApply,
intEquality,
sqequalBase,
inlFormation_alt,
addEquality,
productEquality,
inrFormation_alt,
pointwiseFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[locs,b:bag(T)].
\#(b) \mleq{} \#(locs) supposing bag-no-repeats(T;b) \mwedge{} (\mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} x \mdownarrow{}\mmember{} locs))
Date html generated:
2020_05_20-AM-08_03_04
Last ObjectModification:
2019_11_27-PM-03_05_43
Theory : bags
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