Nuprl Lemma : b_all-map
∀[A,B:Type].
∀f:A ⟶ B. ∀b:bag(A). ∀P:B ⟶ ℙ. ((∀b:B. SqStable(P[b])) ⇒ (b_all(B;bag-map(f;b);x.P[x]) ⇐⇒ b_all(A;b;x.P[f x])))
Proof
Definitions occuring in Statement :
b_all: b_all(T;b;x.P[x]),
bag-map: bag-map(f;bs),
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,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
b_all: b_all(T;b;x.P[x]),
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
guard: {T}
Lemmas referenced :
bag-member_wf,
equal_wf,
all_wf,
squash_wf,
exists_wf,
bag-member-map,
bag-map_wf,
iff_wf,
sq_stable_wf,
bag_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
independent_functionElimination,
dependent_pairFormation,
productEquality,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
lambdaEquality,
functionEquality,
addLevel,
productElimination,
impliesFunctionality,
allFunctionality,
independent_isectElimination,
allLevelFunctionality,
impliesLevelFunctionality,
because_Cache,
universeEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A,B:Type].
\mforall{}f:A {}\mrightarrow{} B. \mforall{}b:bag(A). \mforall{}P:B {}\mrightarrow{} \mBbbP{}.
((\mforall{}b:B. SqStable(P[b])) {}\mRightarrow{} (b\_all(B;bag-map(f;b);x.P[x]) \mLeftarrow{}{}\mRightarrow{} b\_all(A;b;x.P[f x])))
Date html generated:
2017_10_01-AM-08_55_13
Last ObjectModification:
2017_07_26-PM-04_37_10
Theory : bags
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