Nuprl Lemma : length-one-iff
∀[T:Type]. ∀[L:T List].
uiff(||L|| = 1 ∈ ℤ;(∀[x,y:T]. (x = y ∈ T) supposing ((y ∈ L) and (x ∈ L))) ∧ no_repeats(T;L) ∧ 0 < ||L||)
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
length: ||as||,
list: T List,
less_than: a < b,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
ge: i ≥ j ,
cand: A c∧ B,
cons: [a / b],
true: True,
guard: {T},
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
select: L[n],
subtract: n - m
Lemmas referenced :
l_member_wf,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
no_repeats_witness,
member-less_than,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
no_repeats_wf,
istype-less_than,
length_wf,
list_wf,
istype-universe,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
no_repeats_cons,
cons_wf,
cons_member,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
select_wf,
nat_properties,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
subtype_base_sq,
no_repeats_nil,
satisfiable-full-omega-tt,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
length-one-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesis,
Error :universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
dependent_functionElimination,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
voidElimination,
productElimination,
independent_pairEquality,
because_Cache,
Error :equalityIstype,
applyEquality,
intEquality,
baseClosed,
sqequalBase,
Error :productIsType,
Error :isectIsType,
instantiate,
universeEquality,
lemma_by_obid,
lambdaFormation,
computeAll,
lambdaEquality,
dependent_pairFormation,
rename,
voidEquality,
isect_memberEquality,
hypothesis_subsumption,
promote_hyp,
cumulativity,
imageElimination,
Error :inlFormation_alt,
Error :dependent_set_memberEquality_alt,
Error :lambdaFormation_alt,
setElimination,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List].
uiff(||L|| = 1;(\mforall{}[x,y:T]. (x = y) supposing ((y \mmember{} L) and (x \mmember{} L))) \mwedge{} no\_repeats(T;L) \mwedge{} 0 < ||L||)
Date html generated:
2019_06_20-PM-01_27_33
Last ObjectModification:
2019_03_06-AM-11_18_26
Theory : list_1
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