### Nuprl Lemma : bag-rep-size-restrict

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[b:bag(T)].  (bag-rep(#((b|x));x) = (b|x) ∈ bag(T))`

Proof

Definitions occuring in Statement :  bag-restrict: `(b|x)` bag-rep: `bag-rep(n;x)` bag-size: `#(bs)` bag: `bag(T)` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` squash: `↓T` exists: `∃x:A. B[x]` bag-restrict: `(b|x)` bag-size: `#(bs)` bag-rep: `bag-rep(n;x)` bag-filter: `[x∈b|p[x]]` so_lambda: `λ2x.t[x]` all: `∀x:A. B[x]` prop: `ℙ` deq: `EqDecider(T)` subtype_rel: `A ⊆r B` uimplies: `b supposing a` so_apply: `x[s]` implies: `P `` Q` top: `Top` empty-bag: `{}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` eqof: `eqof(d)` ifthenelse: `if b then t else f fi ` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` ge: `i ≥ j ` le: `A ≤ B` satisfiable_int_formula: `satisfiable_int_formula(fmla)` cons-bag: `x.b` true: `True`
Lemmas referenced :  bag_to_squash_list list_induction equal_wf bag_wf primrec_wf length_wf_nat filter_wf5 l_member_wf empty-bag_wf cons-bag_wf int_seg_wf length_wf list-subtype-bag list_wf filter_nil_lemma length_of_nil_lemma primrec0_lemma nil_wf filter_cons_lemma bag-rep_wf bag-size_wf bag-restrict_wf deq_wf bool_wf eqtt_to_assert safe-assert-deq length_of_cons_lemma primrec-unroll eq_int_wf assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int non_neg_length eqof_wf satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_formula_prop_wf squash_wf true_wf add-subtract-cancel
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality imageElimination productElimination promote_hyp hypothesis rename sqequalRule lambdaEquality cumulativity lambdaFormation setElimination applyEquality setEquality natural_numberEquality independent_isectElimination independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality hyp_replacement equalitySymmetry applyLambdaEquality axiomEquality universeEquality unionElimination equalityElimination equalityTransitivity addEquality dependent_pairFormation instantiate int_eqEquality intEquality independent_pairFormation computeAll imageMemberEquality baseClosed

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[x:T].  \mforall{}[b:bag(T)].    (bag-rep(\#((b|x));x)  =  (b|x))

Date html generated: 2018_05_21-PM-09_52_42
Last ObjectModification: 2017_07_26-PM-06_32_01

Theory : bags_2

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