Nuprl Lemma : sub-bag-admissable
∀[T:Type]. ∀[R:bag(T) ⟶ bag(T) ⟶ ℙ].
(bag-admissable(T;as,bs.R[as;bs]) ⇒ (∀as,bs:bag(T). (sub-bag(T;as;bs) ⇒ R[as;bs])))
Proof
Definitions occuring in Statement :
bag-admissable: bag-admissable(T;as,bs.R[as; bs]),
sub-bag: sub-bag(T;as;bs),
bag: bag(T),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
sub-bag: sub-bag(T;as;bs),
exists: ∃x:A. B[x],
bag-admissable: bag-admissable(T;as,bs.R[as; bs]),
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
guard: {T},
top: Top,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
bag_wf,
sub-bag_wf,
bag-admissable_wf,
empty-bag_wf,
bag-empty-append,
squash_wf,
true_wf,
bag-append-comm,
bag-append_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
equalityTransitivity,
applyEquality,
functionExtensionality,
sqequalRule,
lambdaEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:bag(T) {}\mrightarrow{} bag(T) {}\mrightarrow{} \mBbbP{}].
(bag-admissable(T;as,bs.R[as;bs]) {}\mRightarrow{} (\mforall{}as,bs:bag(T). (sub-bag(T;as;bs) {}\mRightarrow{} R[as;bs])))
Date html generated:
2017_10_01-AM-09_05_09
Last ObjectModification:
2017_07_26-PM-04_45_06
Theory : bags
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