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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