Nuprl Lemma : b_all-map2
∀[A,B:Type].
∀b:bag(A). ∀f:{a:A| a ↓∈ b} ⟶ B. ∀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-member: x ↓∈ bs,
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,
set: {x:A| B[x]} ,
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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
squash: ↓T,
exists: ∃x:A. B[x],
uimplies: b supposing a,
uiff: uiff(P;Q),
b_all: b_all(T;b;x.P[x]),
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
b_all_wf,
bag-map_wf,
bag-member_wf,
bag-subtype,
istype-universe,
sq_stable_wf,
bag_wf,
equal_wf,
bag-member-map2,
sq_stable__bag-member,
subtype_rel_self,
iff_weakening_equal,
squash_wf,
exists_wf,
iff_weakening_uiff,
b_all-wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
setEquality,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
lambdaEquality_alt,
applyEquality,
functionIsType,
universeEquality,
setIsType,
inhabitedIsType,
baseClosed,
imageMemberEquality,
rename,
setElimination,
dependent_pairFormation,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
lambdaFormation,
imageElimination,
instantiate,
dependent_set_memberEquality_alt
Latex:
\mforall{}[A,B:Type].
\mforall{}b:bag(A). \mforall{}f:\{a:A| a \mdownarrow{}\mmember{} b\} {}\mrightarrow{} B. \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:
2019_10_15-AM-11_02_51
Last ObjectModification:
2018_10_12-AM-09_52_59
Theory : bags
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