Nuprl Lemma : no_repeats_cons
∀[T:Type]. ∀[l:T List]. ∀[x:T]. uiff(no_repeats(T;[x / l]);no_repeats(T;l) ∧ (¬(x ∈ l)))
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
no_repeats: no_repeats(T;l),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
guard: {T},
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
select: L[n],
cons: [a / b],
sq_type: SQType(T),
label: ...$L... t
Lemmas referenced :
length_of_cons_lemma,
istype-void,
select_wf,
sq_stable__le,
not_wf,
equal_wf,
nat_wf,
less_than_wf,
length_wf,
l_member_wf,
cons_wf,
list_wf,
le_antisymmetry_iff,
not-equal-2,
decidable__equal_int,
le-add-cancel2,
less-iff-le,
not-lt-2,
decidable__lt,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
false_wf,
decidable__le,
squash_wf,
true_wf,
select_cons_tl,
iff_weakening_equal,
istype-false,
istype-int,
int_subtype_base,
subtract_wf,
minus-minus,
subtype_rel_self,
minus-zero,
le-add-cancel-alt,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :isect_memberEquality_alt,
voidElimination,
hypothesis,
Error :isect_memberFormation_alt,
independent_pairFormation,
Error :lambdaFormation_alt,
independent_functionElimination,
Error :equalityIsType1,
Error :inhabitedIsType,
hypothesisEquality,
isectElimination,
cumulativity,
setElimination,
rename,
because_Cache,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
Error :lambdaEquality_alt,
Error :universeIsType,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_pairEquality,
Error :isectIsType,
addEquality,
Error :productIsType,
universeEquality,
applyLambdaEquality,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
applyEquality,
lambdaFormation,
unionElimination,
dependent_set_memberEquality,
Error :dependent_set_memberEquality_alt,
Error :equalityIsType4,
instantiate,
Error :dependent_pairFormation_alt,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[l:T List]. \mforall{}[x:T]. uiff(no\_repeats(T;[x / l]);no\_repeats(T;l) \mwedge{} (\mneg{}(x \mmember{} l)))
Date html generated:
2019_06_20-PM-00_42_15
Last ObjectModification:
2018_10_01-PM-10_15_57
Theory : list_0
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