### Nuprl Lemma : bag-maximal?-append

`∀[T:Type]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[b1,b2:bag(T)]. ∀[x:T].`
`  uiff(↑bag-maximal?(b1 + b2;x;R);(↑bag-maximal?(b1;x;R)) ∧ (↑bag-maximal?(b2;x;R)))`

Proof

Definitions occuring in Statement :  bag-maximal?: `bag-maximal?(bg;x;R)` bag-append: `as + bs` bag: `bag(T)` assert: `↑b` bool: `𝔹` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` prop: `ℙ` and: `P ∧ Q` subtype_rel: `A ⊆r B` uimplies: `b supposing a` bag-maximal?: `bag-maximal?(bg;x;R)` bag-append: `as + bs` bag-accum: `bag-accum(v,x.f[v; x];init;bs)` top: `Top` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` iff: `P `⇐⇒` Q` implies: `P `` Q` true: `True` rev_implies: `P `` Q` sq_type: `SQType(T)` guard: `{T}` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` bfalse: `ff` not: `¬A` false: `False` so_apply: `x[s]` band: `p ∧b q` squash: `↓T` sq_stable: `SqStable(P)` exists: `∃x:A. B[x]`
Lemmas referenced :  assert_wf bag-maximal?_wf bag-append_wf squash_wf list-subtype-bag list_accum_append subtype_rel_list top_wf decidable__assert list_accum_wf bool_wf btrue_wf band_wf iff_imp_equal_bool true_wf subtype_base_sq bool_subtype_base assert_functionality_wrt_uiff assert_witness bfalse_wf false_wf list_accum_invariant not_wf assert_elim not_assert_elim and_wf equal_wf btrue_neq_bfalse uiff_wf bag_wf bag_to_squash_list sq_stable__uiff sq_stable_from_decidable sq_stable__and
Rules used in proof :  because_Cache sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity hypothesisEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesis functionExtensionality applyEquality productEquality isect_memberEquality independent_isectElimination lambdaEquality sqequalRule voidElimination voidEquality dependent_functionElimination unionElimination independent_pairFormation lambdaFormation natural_numberEquality instantiate equalityTransitivity equalitySymmetry independent_functionElimination isect_memberFormation productElimination independent_pairEquality addLevel levelHypothesis dependent_set_memberEquality setElimination rename setEquality functionEquality universeEquality imageElimination promote_hyp hyp_replacement Error :applyLambdaEquality,  imageMemberEquality baseClosed

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[b1,b2:bag(T)].  \mforall{}[x:T].
uiff(\muparrow{}bag-maximal?(b1  +  b2;x;R);(\muparrow{}bag-maximal?(b1;x;R))  \mwedge{}  (\muparrow{}bag-maximal?(b2;x;R)))

Date html generated: 2016_10_25-AM-10_24_33
Last ObjectModification: 2016_07_12-AM-06_41_10

Theory : bags

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