### Nuprl Lemma : cardinality-le-list

`∀[T:Type]. ∀n:ℕ. (|T| ≤ n `` (∃L:T List. ((||L|| = n ∈ ℤ) ∧ (∀x:T. (x ∈ L)))))`

Proof

Definitions occuring in Statement :  cardinality-le: `|T| ≤ n` l_member: `(x ∈ l)` length: `||as||` list: `T List` nat: `ℕ` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  cardinality-le: `|T| ≤ n` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` prop: `ℙ` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` and: `P ∧ Q` cand: `A c∧ B` top: `Top` surject: `Surj(A;B;f)` l_member: `(x ∈ l)` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` guard: `{T}` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  exists_wf int_seg_wf surject_wf nat_wf mklist_wf mklist_length equal_wf length_wf all_wf l_member_wf int_seg_subtype_nat false_wf int_seg_properties nat_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf less_than_wf select_wf decidable__le intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma squash_wf true_wf mklist_select iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin cut introduction extract_by_obid isectElimination functionEquality natural_numberEquality setElimination rename hypothesisEquality hypothesis cumulativity lambdaEquality because_Cache functionExtensionality applyEquality universeEquality dependent_pairFormation isect_memberEquality voidElimination voidEquality independent_pairFormation productEquality intEquality dependent_functionElimination equalityTransitivity equalitySymmetry independent_isectElimination unionElimination int_eqEquality computeAll imageElimination imageMemberEquality baseClosed independent_functionElimination

Latex:
\mforall{}[T:Type].  \mforall{}n:\mBbbN{}.  (|T|  \mleq{}  n  {}\mRightarrow{}  (\mexists{}L:T  List.  ((||L||  =  n)  \mwedge{}  (\mforall{}x:T.  (x  \mmember{}  L)))))

Date html generated: 2017_04_17-AM-07_45_16
Last ObjectModification: 2017_02_27-PM-04_17_04

Theory : list_1

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