### Nuprl Lemma : nth_tl_is_fseg

`∀[T:Type]. ∀L1,L2:T List.  (fseg(T;L1;L2) `⇐⇒` ∃n:ℕ||L2|| + 1. (L1 = nth_tl(n;L2) ∈ (T List)))`

Proof

Definitions occuring in Statement :  fseg: `fseg(T;L1;L2)` length: `||as||` nth_tl: `nth_tl(n;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 :  fseg: `fseg(T;L1;L2)` 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` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` squash: `↓T` true: `True` guard: `{T}` less_than: `a < b` uiff: `uiff(P;Q)` int_iseg: `{i...j}`
Lemmas referenced :  exists_wf list_wf equal_wf append_wf int_seg_wf length_wf nth_tl_wf non_neg_length 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 lelt_wf nth_tl_append int_seg_subtype false_wf le_wf squash_wf true_wf add_functionality_wrt_eq length_append subtype_rel_list top_wf iff_weakening_equal int_seg_properties add-is-int-iff firstn_wf append_firstn_lastn subtype_rel_sets
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin cut introduction extract_by_obid isectElimination cumulativity hypothesisEquality hypothesis lambdaEquality natural_numberEquality addEquality setElimination rename universeEquality dependent_pairFormation dependent_set_memberEquality dependent_functionElimination because_Cache unionElimination independent_isectElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll addLevel applyEquality imageElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed independent_functionElimination applyLambdaEquality pointwiseFunctionality promote_hyp baseApply closedConclusion hyp_replacement equalityUniverse levelHypothesis productEquality setEquality

Latex:
\mforall{}[T:Type].  \mforall{}L1,L2:T  List.    (fseg(T;L1;L2)  \mLeftarrow{}{}\mRightarrow{}  \mexists{}n:\mBbbN{}||L2||  +  1.  (L1  =  nth\_tl(n;L2)))

Date html generated: 2017_04_17-AM-07_33_05
Last ObjectModification: 2017_02_27-PM-04_09_23

Theory : list_1

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