### Nuprl Lemma : frs-increasing-separated-common-refinement

`∀p,q:ℝ List.`
`  (frs-increasing(p)`
`  `` frs-increasing(q)`
`  `` frs-separated(p;q)`
`  `` (∃r:ℝ List. (frs-increasing(r) ∧ frs-refines(r;p) ∧ frs-refines(r;q) ∧ frs-refines(p @ q;r))))`

Proof

Definitions occuring in Statement :  frs-separated: `frs-separated(p;q)` frs-increasing: `frs-increasing(p)` frs-refines: `frs-refines(p;q)` real: `ℝ` append: `as @ bs` list: `T List` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x y.t[x; y]` trans: `Trans(T;x,y.E[x; y])` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` and: `P ∧ Q` frs-separated: `frs-separated(p;q)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rneq: `x ≠ y` exists: `∃x:A. B[x]` cand: `A c∧ B` frs-refines: `frs-refines(p;q)` l_all: `(∀x∈L.P[x])` l_contains: `A ⊆ B` l_member: `(x ∈ l)` l_exists: `(∃x∈L. P[x])` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` 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` uiff: `uiff(P;Q)`
Lemmas referenced :  merge-strict-exists real_wf rless_wf rless_transitivity2 rleq_weakening_rless l_member_wf frs-increasing-sorted-by frs-separated_wf frs-increasing_wf list_wf l_all_iff l_all_wf2 rneq_wf all_wf lelt_wf length_wf req_weakening req_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 int_formula_prop_less_lemma int_seg_wf length-append append_wf add-is-int-iff itermAdd_wf int_term_value_add_lemma false_wf frs-refines_wf le_wf and_wf equal_wf nat_wf less_than_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis sqequalRule lambdaEquality hypothesisEquality independent_functionElimination dependent_functionElimination independent_isectElimination because_Cache productElimination independent_pairFormation addLevel setElimination rename setEquality allFunctionality levelHypothesis promote_hyp functionEquality dependent_pairFormation dependent_set_memberEquality equalityTransitivity equalitySymmetry natural_numberEquality unionElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll imageElimination addEquality pointwiseFunctionality baseApply closedConclusion baseClosed productEquality hyp_replacement applyEquality

Latex:
\mforall{}p,q:\mBbbR{}  List.
(frs-increasing(p)
{}\mRightarrow{}  frs-increasing(q)
{}\mRightarrow{}  frs-separated(p;q)
{}\mRightarrow{}  (\mexists{}r:\mBbbR{}  List.  (frs-increasing(r)  \mwedge{}  frs-refines(r;p)  \mwedge{}  frs-refines(r;q)  \mwedge{}  frs-refines(p  @  q;r))))

Date html generated: 2016_10_26-AM-09_33_11
Last ObjectModification: 2016_07_12-AM-08_20_23

Theory : reals

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