### Nuprl Lemma : bag-decomp_wf

`∀[T:Type]. ∀[bs:bag(T)].  (bag-decomp(bs) ∈ bag(T × bag(T)))`

Proof

Definitions occuring in Statement :  bag-decomp: `bag-decomp(bs)` bag: `bag(T)` uall: `∀[x:A]. B[x]` member: `t ∈ T` product: `x:A × B[x]` universe: `Type`
Definitions unfolded in proof :  prop: `ℙ` implies: `P `` Q` so_apply: `x[s]` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` uimplies: `b supposing a` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` bag-decomp: `bag-decomp(bs)` and: `P ∧ Q` quotient: `x,y:A//B[x; y]` bag: `bag(T)` member: `t ∈ T` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` permutation: `permutation(T;L1;L2)` ge: `i ≥ j ` nat: `ℕ` sq_type: `SQType(T)` false: `False` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` or: `P ∨ Q` decidable: `Dec(P)` less_than: `a < b` le: `A ≤ B` lelt: `i ≤ j < k` int_seg: `{i..j-}` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` guard: `{T}` true: `True` squash: `↓T` remove-nth: `remove-nth(n;L)` int_iseg: `{i...j}` cand: `A c∧ B` less_than': `less_than'(a;b)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` inject: `Inj(A;B;f)` subtract: `n - m` respects-equality: `respects-equality(S;T)`
Rules used in proof :  universeEquality instantiate isectIsTypeImplies isect_memberEquality_alt equalityTransitivity axiomEquality equalitySymmetry sqequalBase equalityIstype productIsType independent_functionElimination lambdaFormation_alt applyEquality natural_numberEquality dependent_functionElimination because_Cache independent_isectElimination universeIsType inhabitedIsType lambdaEquality_alt productElimination promote_hyp pertypeElimination sqequalRule hypothesis hypothesisEquality productEquality thin isectElimination extract_by_obid pointwiseFunctionalityForEquality sqequalHypSubstitution cut introduction isect_memberFormation_alt sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution dependent_pairFormation_alt Error :memTop,  independent_pairFormation applyLambdaEquality hyp_replacement closedConclusion baseApply dependent_set_memberEquality_alt cumulativity voidElimination int_eqEquality approximateComputation unionElimination rename setElimination baseClosed imageMemberEquality intEquality imageElimination functionIsType independent_pairEquality addEquality pointwiseFunctionality equalityElimination functionEquality minusEquality multiplyEquality functionExtensionality

Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].    (bag-decomp(bs)  \mmember{}  bag(T  \mtimes{}  bag(T)))

Date html generated: 2020_05_20-AM-08_02_56
Last ObjectModification: 2020_01_31-PM-03_36_11

Theory : bags

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