### Nuprl Lemma : member-firstn

`∀[T:Type]. ∀L:T List. ∀n:ℕ||L|| + 1. ∀x:T.  ((x ∈ firstn(n;L)) `⇐⇒` ∃i:ℕn. (x = L[i] ∈ T))`

Proof

Definitions occuring in Statement :  firstn: `firstn(n;as)` l_member: `(x ∈ l)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` iff: `P `⇐⇒` Q` exists: `∃x:A. B[x]` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` rev_implies: `P `` Q` guard: `{T}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` less_than: `a < b` squash: `↓T` cand: `A c∧ B` so_lambda: `λ2x.t[x]` uiff: `uiff(P;Q)` so_apply: `x[s]`
Lemmas referenced :  member_firstn int_seg_subtype_nat length_wf false_wf lelt_wf equal_wf select_wf int_seg_properties nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma l_member_wf firstn_wf less_than_wf exists_wf int_seg_wf add-is-int-iff list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache dependent_functionElimination hypothesisEquality applyEquality natural_numberEquality addEquality cumulativity hypothesis independent_isectElimination sqequalRule independent_pairFormation productElimination independent_functionElimination dependent_pairFormation setElimination rename dependent_set_memberEquality unionElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll imageElimination productEquality pointwiseFunctionality equalityTransitivity equalitySymmetry promote_hyp baseApply closedConclusion baseClosed universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}n:\mBbbN{}||L||  +  1.  \mforall{}x:T.    ((x  \mmember{}  firstn(n;L))  \mLeftarrow{}{}\mRightarrow{}  \mexists{}i:\mBbbN{}n.  (x  =  L[i]))

Date html generated: 2017_04_17-AM-07_51_53
Last ObjectModification: 2017_02_27-PM-04_25_01

Theory : list_1

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