Nuprl Lemma : blended-real-req

[k:ℕ+]. ∀[x,y:ℝ].  blended-real(k;x;y) supposing |x y| ≤ (r1/r(k))


Proof




Definitions occuring in Statement :  blended-real: blended-real(k;x;y) rdiv: (x/y) rleq: x ≤ y rabs: |x| rsub: y req: y int-to-real: r(n) real: nat_plus: + uimplies: supposing a uall: [x:A]. B[x] natural_number: $n
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] nat_plus: + less_than: a < b squash: T less_than': less_than'(a;b) true: True and: P ∧ Q prop: cand: c∧ B real: uiff: uiff(P;Q) guard: {T} implies:  Q rneq: x ≠ y or: P ∨ Q iff: ⇐⇒ Q rev_implies:  Q decidable: Dec(P) not: ¬A satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False top: Top subtype_rel: A ⊆B so_lambda: λ2x.t[x] int_upper: {i...} le: A ≤ B so_apply: x[s] accelerate: accelerate(k;f) blended-real: blended-real(k;x;y) blend-seq: blend-seq(k;x;y) has-value: (a)↓ bool: 𝔹 unit: Unit it: btrue: tt ifthenelse: if then else fi  bfalse: ff sq_type: SQType(T) bnot: ¬bb assert: b nequal: a ≠ b ∈ 
Lemmas referenced :  real-regular less_than_wf req-iff-bdd-diff blended-real_wf accelerate_wf regular-int-seq_wf nat_plus_wf accelerate-bdd-diff req_transitivity req_witness rleq_wf rabs_wf rsub_wf rdiv_wf int-to-real_wf rless-int nat_plus_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_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_wf rless_wf real_wf eventually-equal-implies-bdd-diff int_upper_wf all_wf equal_wf less_than_transitivity1 value-type-has-value int-value-type lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int int_upper_properties itermMultiply_wf intformle_wf int_term_value_mul_lemma int_formula_prop_le_lemma eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot false_wf not-lt-2 less-iff-le add_functionality_wrt_le add-associates add-zero add-swap add-commutes zero-add le-add-cancel int_subtype_base equal-wf-base true_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_set_memberEquality natural_numberEquality sqequalRule independent_pairFormation imageMemberEquality baseClosed hypothesis because_Cache independent_isectElimination setElimination rename dependent_functionElimination functionExtensionality applyEquality productElimination independent_functionElimination inrFormation unionElimination approximateComputation dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality equalityTransitivity equalitySymmetry callbyvalueReduce sqleReflexivity multiplyEquality equalityElimination promote_hyp instantiate cumulativity divideEquality addEquality addLevel

Latex:
\mforall{}[k:\mBbbN{}\msupplus{}].  \mforall{}[x,y:\mBbbR{}].    blended-real(k;x;y)  =  y  supposing  |x  -  y|  \mleq{}  (r1/r(k))



Date html generated: 2017_10_03-AM-10_08_56
Last ObjectModification: 2017_07_05-PM-04_27_51

Theory : reals


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