Nuprl Lemma : bag-maximals-max

[T:Type]. ∀[b:bag(T)]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[x,y:T].  (↑(R y)) supposing (x ↓∈ bag-maximals(b;R) and y ↓∈ b)


Definitions occuring in Statement :  bag-member: x ↓∈ bs bag-maximals: bag-maximals(bg;R) bag: bag(T) assert: b bool: 𝔹 uimplies: supposing a uall: [x:A]. B[x] apply: a function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  member: t ∈ T uall: [x:A]. B[x] bag-maximals: bag-maximals(bg;R) so_lambda: λ2x.t[x] subtype_rel: A ⊆B uimplies: supposing a so_apply: x[s] uiff: uiff(P;Q) and: P ∧ Q implies:  Q prop: squash: T exists: x:A. B[x] all: x:A. B[x] sq_stable: SqStable(P)
Lemmas referenced :  bag-member_wf bag-maximals_wf bag-member-filter bag-maximal?_wf list-subtype-bag assert_wf bag-maximal?-max bool_wf bag_wf assert_witness bag_to_squash_list sq_stable_from_decidable decidable__assert
Rules used in proof :  because_Cache sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity hypothesisEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity functionExtensionality applyEquality hypothesis sqequalRule lambdaEquality independent_isectElimination productElimination functionEquality universeEquality isect_memberFormation independent_functionElimination isect_memberEquality equalityTransitivity equalitySymmetry imageElimination promote_hyp hyp_replacement Error :applyLambdaEquality,  rename dependent_functionElimination imageMemberEquality baseClosed

\mforall{}[T:Type].  \mforall{}[b:bag(T)].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[x,y:T].
    (\muparrow{}(R  x  y))  supposing  (x  \mdownarrow{}\mmember{}  bag-maximals(b;R)  and  y  \mdownarrow{}\mmember{}  b)

Date html generated: 2016_10_25-AM-10_32_06
Last ObjectModification: 2016_07_12-AM-06_47_35

Theory : bags

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