### Nuprl Lemma : nth_tl-partition

`∀I:Interval`
`  (icompact(I)`
`  `` (∀a:ℝ. ∀p:partition(I). ∀i:ℕ||p||.  ((a = p[i]) `` (nth_tl(i + 1;p) ∈ partition([a, right-endpoint(I)])))))`

Proof

Definitions occuring in Statement :  partition: `partition(I)` icompact: `icompact(I)` rccint: `[l, u]` right-endpoint: `right-endpoint(I)` interval: `Interval` req: `x = y` real: `ℝ` select: `L[n]` length: `||as||` nth_tl: `nth_tl(n;as)` int_seg: `{i..j-}` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` partition: `partition(I)` uall: `∀[x:A]. B[x]` int_seg: `{i..j-}` partitions: `partitions(I;p)` and: `P ∧ Q` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` iff: `P `⇐⇒` Q` icompact: `icompact(I)` cand: `A c∧ B` frs-non-dec: `frs-non-dec(L)` int_iseg: `{i...j}` true: `True` subtype_rel: `A ⊆r B` le: `A ≤ B` uiff: `uiff(P;Q)` rev_implies: `P `` Q` left-endpoint: `left-endpoint(I)` endpoints: `endpoints(I)` rccint: `[l, u]` outl: `outl(x)` pi1: `fst(t)` nat: `ℕ` less_than': `less_than'(a;b)` subtract: `n - m` right-endpoint: `right-endpoint(I)` pi2: `snd(t)` last: `last(L)` sq_type: `SQType(T)` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` cons: `[a / b]` bfalse: `ff` sorted-by: `sorted-by(R;L)`
Lemmas referenced :  nth_tl_wf real_wf int_seg_properties length_wf decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_wf partitions_wf rccint_wf right-endpoint_wf rccint-icompact req_wf select_wf decidable__le int_seg_wf partition_wf icompact_wf interval_wf less_than_wf squash_wf true_wf length_nth_tl itermAdd_wf int_term_value_add_lemma le_wf iff_weakening_equal itermSubtract_wf int_term_value_subtract_lemma lelt_wf subtract_wf add-member-int_seg2 rleq_wf select_nth_tl select-nth_tl false_wf subtype_rel_list top_wf add-zero rleq_transitivity rleq_weakening subtype_base_sq int_subtype_base decidable__equal_int intformeq_wf int_formula_prop_eq_lemma equal_wf add_functionality_wrt_eq frs-non-dec-sorted-by list-cases stuck-spread base_wf length_of_nil_lemma null_nil_lemma product_subtype_list length_of_cons_lemma null_cons_lemma last_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut sqequalHypSubstitution setElimination thin rename dependent_set_memberEquality introduction extract_by_obid isectElimination hypothesis hypothesisEquality addEquality natural_numberEquality productElimination independent_functionElimination dependent_functionElimination unionElimination imageElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll because_Cache applyEquality equalityTransitivity equalitySymmetry productEquality imageMemberEquality baseClosed universeEquality instantiate cumulativity promote_hyp hypothesis_subsumption

Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{}  (\mforall{}a:\mBbbR{}.  \mforall{}p:partition(I).  \mforall{}i:\mBbbN{}||p||.
((a  =  p[i])  {}\mRightarrow{}  (nth\_tl(i  +  1;p)  \mmember{}  partition([a,  right-endpoint(I)])))))

Date html generated: 2017_10_03-AM-09_42_48
Last ObjectModification: 2017_07_28-AM-07_57_22

Theory : reals

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