Nuprl Lemma : bag-member-map3-deq
∀[T,U:Type].
∀x:U. ∀bs:bag(T). ∀f:{v:T| v ↓∈ bs} ⟶ U.
(Inj({v:T| v ↓∈ bs} ;U;f) ⇒ (∀x,y:U. Dec(x = y ∈ U)) ⇒ uiff(x ↓∈ bag-map(f;bs);∃v:T. (v ↓∈ bs ∧ (x = (f v) ∈ U)))\000C)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-map: bag-map(f;bs),
bag: bag(T),
inject: Inj(A;B;f),
decidable: Dec(P),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
bag-member: x ↓∈ bs,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
decision: Decision,
top: Top,
inject: Inj(A;B;f),
cand: A c∧ B,
sq_stable: SqStable(P),
label: ...$L... t,
true: True
Lemmas referenced :
bag-member-map3,
bag-member_wf,
bag-map-member-wf,
squash_wf,
exists_wf,
equal_wf,
iff_weakening_uiff,
decidable_wf,
inject_wf,
bag_wf,
single-valued-bag-filter,
dec2bool_wf,
subtype_rel_self,
subtype_rel_union,
not_wf,
top_wf,
dec2bool_decidable,
assert_wf,
bag-member-size,
bag-filter-wf2,
subtype_rel_bag,
bag-member-filter2,
sv-bag-only_wf,
sq_stable__bag-member,
bag-member-sv-bag-only,
sv-bag-only-filter,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
hypothesis,
independent_pairFormation,
imageElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
rename,
universeIsType,
lambdaEquality,
productEquality,
applyEquality,
dependent_set_memberEquality,
productIsType,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionIsType,
inhabitedIsType,
setEquality,
setIsType,
universeEquality,
lambdaFormation,
setElimination,
functionExtensionality,
cumulativity,
functionEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
natural_numberEquality,
instantiate
Latex:
\mforall{}[T,U:Type].
\mforall{}x:U. \mforall{}bs:bag(T). \mforall{}f:\{v:T| v \mdownarrow{}\mmember{} bs\} {}\mrightarrow{} U.
(Inj(\{v:T| v \mdownarrow{}\mmember{} bs\} ;U;f)
{}\mRightarrow{} (\mforall{}x,y:U. Dec(x = y))
{}\mRightarrow{} uiff(x \mdownarrow{}\mmember{} bag-map(f;bs);\mexists{}v:T. (v \mdownarrow{}\mmember{} bs \mwedge{} (x = (f v)))))
Date html generated:
2019_10_15-AM-11_02_44
Last ObjectModification:
2018_09_27-AM-11_19_21
Theory : bags
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