Nuprl Lemma : b_all-cons
∀[B:Type]. ∀b:bag(B). ∀P:B ⟶ ℙ. ∀x:B. ((∀b:B. SqStable(P[b])) ⇒ (b_all(B;x.b;x.P[x]) ⇐⇒ P[x] ∧ b_all(B;b;x.P[x])))
Proof
Definitions occuring in Statement :
b_all: b_all(T;b;x.P[x]),
cons-bag: x.b,
bag: bag(T),
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
b_all: b_all(T;b;x.P[x]),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
sq_or: a ↓∨ b,
or: P ∨ Q,
squash: ↓T,
sq_stable: SqStable(P),
guard: {T}
Lemmas referenced :
b_all_wf,
cons-bag_wf,
all_wf,
sq_stable_wf,
bag_wf,
bag-member-cons,
bag-member_wf,
sq_stable__bag-member,
equal_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
productElimination,
productEquality,
universeEquality,
functionEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
independent_isectElimination,
inlFormation,
imageMemberEquality,
baseClosed,
inrFormation,
imageElimination,
unionElimination,
equalitySymmetry,
dependent_set_memberEquality,
setElimination,
rename,
setEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[B:Type]
\mforall{}b:bag(B). \mforall{}P:B {}\mrightarrow{} \mBbbP{}. \mforall{}x:B.
((\mforall{}b:B. SqStable(P[b])) {}\mRightarrow{} (b\_all(B;x.b;x.P[x]) \mLeftarrow{}{}\mRightarrow{} P[x] \mwedge{} b\_all(B;b;x.P[x])))
Date html generated:
2016_10_25-AM-10_29_01
Last ObjectModification:
2016_07_12-AM-06_45_05
Theory : bags
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