Nuprl Lemma : no_repeats_inject
∀[T:Type]. ∀[l:T List]. uiff(no_repeats(T;l);Inj(ℕ||l||;T;λi.l[i]))
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
select: L[n],
length: ||as||,
list: T List,
inject: Inj(A;B;f),
int_seg: {i..j-},
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
no_repeats: no_repeats(T;l),
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
not: ¬A,
false: False,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
equal_wf,
select_wf,
sq_stable__le,
int_seg_wf,
length_wf,
not_wf,
nat_wf,
less_than_wf,
no_repeats_witness,
no_repeats_wf,
inject_wf,
list_wf,
decidable__equal_int_seg,
int_seg_subtype_nat,
false_wf,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
lambdaFormation,
sqequalRule,
hypothesis,
extract_by_obid,
isectElimination,
thin,
because_Cache,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
hypothesisEquality,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
cumulativity,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
voidElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
universeEquality,
unionElimination,
applyEquality,
applyLambdaEquality,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[l:T List]. uiff(no\_repeats(T;l);Inj(\mBbbN{}||l||;T;\mlambda{}i.l[i]))
Date html generated:
2017_04_14-AM-08_39_51
Last ObjectModification:
2017_02_27-PM-03_30_12
Theory : list_0
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