### Nuprl Lemma : bag-parts'-no-repeats

`∀[T:Type]`
`  ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)].  bag-no-repeats(bag(T) List+;bag-parts'(eq;bs;x)) supposing valueall-type(T)`

Proof

Definitions occuring in Statement :  bag-parts': `bag-parts'(eq;bs;x)` bag-no-repeats: `bag-no-repeats(T;bs)` bag: `bag(T)` listp: `A List+` deq: `EqDecider(T)` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` bag-no-repeats: `bag-no-repeats(T;bs)` squash: `↓T` bag-parts': `bag-parts'(eq;bs;x)` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` listp: `A List+` so_lambda: `λ2x.t[x]` so_apply: `x[s]` callbyvalueall: callbyvalueall has-value: `(a)↓` has-valueall: `has-valueall(a)` subtype_rel: `A ⊆r B` nat: `ℕ` cand: `A c∧ B` inject: `Inj(A;B;f)` ge: `i ≥ j ` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` cons: `[a / b]` top: `Top` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m`
Lemmas referenced :  bag_wf deq_wf valueall-type_wf bag-null_wf bool_wf eqtt_to_assert assert-bag-null bag-single-no-repeats listp_wf cons_wf_listp nil_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot equal-wf-T-base valueall-type-has-valueall bag-valueall-type set-valueall-type list_wf less_than_wf length_wf list-valueall-type bag-parts_wf evalall-reduce bag-append-no-repeats bag-map_wf empty-bag_wf bag-filter_wf eq_int_wf bag-count_wf hd_wf listp_properties nat_wf subtype_rel_bag assert_wf bag-member_wf bag-map-no-repeats list-cases length_of_nil_lemma product_subtype_list length_of_cons_lemma length_wf_nat decidable__lt false_wf not-lt-2 condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates add-zero le-add-cancel reduce_tl_cons_lemma and_wf tl_wf bag-parts-no-repeats bag-filter-no-repeats bag-member-map bag-member-filter bag-member-parts l_all_iff not_wf l_member_wf cons_member
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis sqequalHypSubstitution imageElimination sqequalRule imageMemberEquality hypothesisEquality thin baseClosed extract_by_obid isectElimination cumulativity isect_memberEquality because_Cache equalityTransitivity equalitySymmetry universeEquality dependent_functionElimination lambdaFormation unionElimination equalityElimination productElimination independent_isectElimination dependent_pairFormation promote_hyp instantiate independent_functionElimination voidElimination lambdaEquality natural_numberEquality callbyvalueReduce setElimination rename applyEquality setEquality independent_pairFormation hypothesis_subsumption voidEquality addEquality intEquality minusEquality dependent_set_memberEquality applyLambdaEquality inlFormation hyp_replacement

Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)].  \mforall{}[x:T].  \mforall{}[bs:bag(T)].    bag-no-repeats(bag(T)  List\msupplus{};bag-parts'(eq;bs;x))
supposing  valueall-type(T)

Date html generated: 2018_05_21-PM-09_51_34
Last ObjectModification: 2017_07_26-PM-06_31_32

Theory : bags_2

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