### Nuprl Lemma : bag-no-repeats-settype

`∀[T:Type]. ∀[bs:bag(T)].  uiff(bag-no-repeats({x:T| x ↓∈ bs} ;bs);bag-no-repeats(T;bs)) supposing ∀x,y:T.  Dec(x = y ∈ T\000C)`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-no-repeats: `bag-no-repeats(T;bs)` bag: `bag(T)` decidable: `Dec(P)` uiff: `uiff(P;Q)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` uiff: `uiff(P;Q)` and: `P ∧ Q` bag-no-repeats: `bag-no-repeats(T;bs)` squash: `↓T` prop: `ℙ` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` cand: `A c∧ B` guard: `{T}` implies: `P `` Q` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` less_than: `a < b`
Lemmas referenced :  bag-no-repeats_wf bag-member_wf bag-subtype all_wf decidable_wf equal_wf bag_wf subtype_rel_list subtype_rel_bag equal_functionality_wrt_subtype_rel2 no_repeats-settype list-subtype-bag no_repeats_wf subtype_rel_self list-set-type2 bag-member-select select_wf int_seg_properties length_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_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_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma int_seg_wf no_repeats-subtype bag-settype and_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation sqequalRule sqequalHypSubstitution imageElimination hypothesis imageMemberEquality hypothesisEquality thin baseClosed extract_by_obid isectElimination setEquality cumulativity dependent_functionElimination equalityTransitivity equalitySymmetry productElimination independent_pairEquality isect_memberEquality because_Cache lambdaEquality universeEquality dependent_pairFormation applyEquality independent_isectElimination setElimination rename independent_functionElimination productEquality lambdaFormation hyp_replacement applyLambdaEquality natural_numberEquality unionElimination int_eqEquality intEquality voidElimination voidEquality computeAll dependent_set_memberEquality addLevel levelHypothesis

Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].    uiff(bag-no-repeats(\{x:T|  x  \mdownarrow{}\mmember{}  bs\}  ;bs);bag-no-repeats(T;bs))  supposing  \mforall{}x\000C,y:T.    Dec(x  =  y)

Date html generated: 2017_10_01-AM-09_01_12
Last ObjectModification: 2017_07_26-PM-04_42_21

Theory : bags

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