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||)


Definitions occuring in Statement :  no_repeats: no_repeats(T;l) l_member: (x ∈ l) length: ||as|| list: List less_than: a < b uiff: uiff(P;Q) uimplies: supposing a uall: [x:A]. B[x] and: P ∧ Q natural_number: $n int: universe: Type equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uiff: uiff(P;Q) and: P ∧ Q uimplies: supposing a prop: all: x:A. B[x] decidable: Dec(P) or: P ∨ Q not: ¬A implies:  Q satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False top: Top subtype_rel: A ⊆B nat: so_lambda: λ2x.t[x] so_apply: x[s] le: A ≤ B ge: i ≥  cand: 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: ⇐⇒ Q rev_implies:  Q l_member: (x ∈ l) select: L[n] subtract: 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

\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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