### Nuprl Lemma : bag-parts'_wf

`∀[T:Type]`
`  ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)].  (bag-parts'(eq;bs;x) ∈ bag(bag(T) List+)) supposing valueall-type(T)`

Proof

Definitions occuring in Statement :  bag-parts': `bag-parts'(eq;bs;x)` bag: `bag(T)` listp: `A List+` deq: `EqDecider(T)` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` member: `t ∈ T` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` 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` iff: `P `⇐⇒` Q` not: `¬A` prop: `ℙ` rev_implies: `P `` Q` 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`
Lemmas referenced :  bag-null_wf bool_wf uiff_transitivity equal-wf-T-base assert_wf bag_wf eqtt_to_assert assert-bag-null single-bag_wf listp_wf cons_wf_listp nil_wf iff_transitivity bnot_wf not_wf iff_weakening_uiff eqff_to_assert assert_of_bnot 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_wf bag-map_wf empty-bag_wf bag-filter_wf eq_int_wf bag-count_wf hd_wf listp_properties nat_wf subtype_rel_bag equal_wf deq_wf valueall-type_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution dependent_functionElimination thin cumulativity hypothesisEquality hypothesis lambdaFormation unionElimination equalityElimination isectElimination equalityTransitivity equalitySymmetry baseClosed independent_functionElimination because_Cache productElimination independent_isectElimination independent_pairFormation impliesFunctionality lambdaEquality natural_numberEquality callbyvalueReduce setElimination rename applyEquality setEquality axiomEquality isect_memberEquality universeEquality

Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)].  \mforall{}[x:T].  \mforall{}[bs:bag(T)].    (bag-parts'(eq;bs;x)  \mmember{}  bag(bag(T)  List\msupplus{}))
supposing  valueall-type(T)

Date html generated: 2018_05_21-PM-09_51_10
Last ObjectModification: 2017_07_26-PM-06_31_28

Theory : bags_2

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