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
Home
Index