### 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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