### Nuprl Lemma : bag-maximals-max

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

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-maximals: `bag-maximals(bg;R)` bag: `bag(T)` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f 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 ⊆r B` uimplies: `b supposing a` so_apply: `x[s]` uiff: `uiff(P;Q)` and: `P ∧ Q` implies: `P `` Q` prop: `ℙ` 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

Latex:
\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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