### Nuprl Lemma : member-bag-remove-repeats

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. ∀[x:T].  uiff(x ↓∈ bs;x ↓∈ bag-remove-repeats(eq;bs))`

Proof

Definitions occuring in Statement :  bag-remove-repeats: `bag-remove-repeats(eq;bs)` bag-member: `x ↓∈ bs` bag: `bag(T)` deq: `EqDecider(T)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` subtype_rel: `A ⊆r B` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` decidable: `Dec(P)` less_than: `a < b` squash: `↓T` true: `True` bag-member: `x ↓∈ bs` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  bag-member-count lt_int_wf bag-count_wf nat_wf bool_wf eqtt_to_assert assert_of_lt_int false_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot less_than_wf satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformnot_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_formula_prop_wf less_than'_wf ifthenelse_wf le_wf decidable__le bag-member_wf bag-remove-repeats_wf uiff_wf bag_wf deq_wf iff_weakening_uiff count-bag-remove-repeats bool_cases iff_transitivity assert_of_bnot
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination thin because_Cache hypothesisEquality independent_pairFormation isect_memberFormation natural_numberEquality cumulativity hypothesis applyEquality lambdaEquality setElimination rename sqequalRule lambdaFormation unionElimination equalityElimination productElimination independent_isectElimination dependent_pairFormation equalityTransitivity equalitySymmetry promote_hyp dependent_functionElimination instantiate independent_functionElimination voidElimination int_eqEquality intEquality isect_memberEquality voidEquality computeAll independent_pairEquality axiomEquality imageElimination universeEquality imageMemberEquality baseClosed addLevel impliesFunctionality

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[bs:bag(T)].  \mforall{}[x:T].    uiff(x  \mdownarrow{}\mmember{}  bs;x  \mdownarrow{}\mmember{}  bag-remove-repeats(eq;bs))

Date html generated: 2018_05_21-PM-09_47_32
Last ObjectModification: 2017_07_26-PM-06_30_15

Theory : bags_2

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