Nuprl Lemma : bag-combine-no-repeats2
∀[T1,T2:Type]. ∀[f:T1 ⟶ bag(T2)]. ∀[bs:bag(T1)].
(bag-no-repeats(T2;⋃x∈bs.f[x])) supposing
(bag-no-repeats(T1;bs) and
((∀x,y:T1. ∀z:T2. (z ↓∈ f[x] ⇒ z ↓∈ f[y] ⇒ (x = y ∈ T1)))
∧ (∀x:T1. bag-no-repeats(T2;f[x]))
∧ (∀x,y:T1. Dec(x = y ∈ T1))
∧ (∀x,y:T2. Dec(x = y ∈ T2))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag-combine: ⋃x∈bs.f[x],
bag: bag(T),
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x]
Lemmas referenced :
deq-exists,
bag-combine-no-repeats,
bag-no-repeats_wf,
all_wf,
bag-member_wf,
equal_wf,
decidable_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
because_Cache,
independent_isectElimination,
independent_pairFormation,
sqequalRule,
imageElimination,
imageMemberEquality,
baseClosed,
cumulativity,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
universeEquality
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[f:T1 {}\mrightarrow{} bag(T2)]. \mforall{}[bs:bag(T1)].
(bag-no-repeats(T2;\mcup{}x\mmember{}bs.f[x])) supposing
(bag-no-repeats(T1;bs) and
((\mforall{}x,y:T1. \mforall{}z:T2. (z \mdownarrow{}\mmember{} f[x] {}\mRightarrow{} z \mdownarrow{}\mmember{} f[y] {}\mRightarrow{} (x = y)))
\mwedge{} (\mforall{}x:T1. bag-no-repeats(T2;f[x]))
\mwedge{} (\mforall{}x,y:T1. Dec(x = y))
\mwedge{} (\mforall{}x,y:T2. Dec(x = y))))
Date html generated:
2018_05_21-PM-09_46_59
Last ObjectModification:
2017_07_26-PM-06_30_04
Theory : bags_2
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