### Nuprl Lemma : l_member-alt-def

`∀[T:Type]. ∀L:T List. ∀x:T.  ((x ∈ L) `⇐⇒` ∃i:ℕ||L||. (x = L[i] ∈ T))`

Proof

Definitions occuring in Statement :  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` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  l_member: `(x ∈ l)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` prop: `ℙ` cand: `A c∧ B` uimplies: `b supposing a` guard: `{T}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` less_than: `a < b` squash: `↓T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)`
Lemmas referenced :  lelt_wf equal_wf select_wf int_seg_properties length_wf 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 int_formula_prop_less_lemma exists_wf nat_wf less_than_wf int_seg_subtype_nat false_wf int_seg_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin dependent_pairFormation setElimination rename dependent_set_memberEquality hypothesisEquality hypothesis cut introduction extract_by_obid isectElimination because_Cache addLevel levelHypothesis independent_isectElimination natural_numberEquality cumulativity dependent_functionElimination unionElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll imageElimination productEquality applyEquality universeEquality

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

Date html generated: 2017_04_17-AM-07_25_28
Last ObjectModification: 2017_02_27-PM-04_04_19

Theory : list_1

Home Index