### 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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