Nuprl Lemma : dd_wf

d:ℕ+. ∀x:ℝ.
  (d decimal digits of x  ∈ {a:Atom| "display-as" ∈ Atom} 
   × {a:Atom| "decimal-rational" ∈ Atom} 
   × {z:ℝx} 
   × {n:ℕ+d ∈ ℤ
   × {n:ℤ|x (r(n)/r(10^d))| ≤ (r(2)/r(10^d))} )


Definitions occuring in Statement :  dd: decimal digits of  rdiv: (x/y) rleq: x ≤ y rabs: |x| rsub: y req: y int-to-real: r(n) real: fastexp: i^n nat_plus: + all: x:A. B[x] member: t ∈ T set: {x:A| B[x]}  product: x:A × B[x] natural_number: $n int: token: "$token" atom: Atom equal: t ∈ T
Definitions unfolded in proof :  all: x:A. B[x] member: t ∈ T dd: decimal digits of  uall: [x:A]. B[x] nat_plus: + less_than: a < b squash: T less_than': less_than'(a;b) true: True and: P ∧ Q implies:  Q has-value: (a)↓ uimplies: supposing a so_lambda: λ2x.t[x] so_apply: x[s] subtype_rel: A ⊆B sq_type: SQType(T) guard: {T} exp: i^n primrec: primrec(n;b;c) subtract: m nat: decidable: Dec(P) or: P ∨ Q not: ¬A satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False top: Top prop: le: A ≤ B nequal: a ≠ b ∈  int_nzero: -o iff: ⇐⇒ Q rev_implies:  Q uiff: uiff(P;Q) rational-approx: (x within 1/n) real: rneq: x ≠ y rev_uimplies: rev_uimplies(P;Q) rge: x ≥ y callbyvalueall: callbyvalueall has-valueall: has-valueall(a)
Lemmas referenced :  exp-fastexp exp_wf_nat_plus istype-less_than real_wf nat_plus_wf value-type-has-value set-value-type less_than_wf istype-int int-value-type nat_plus_subtype_nat subtype_base_sq set_subtype_base int_subtype_base nat_plus_properties primrec-wf-nat-plus equal-wf-base decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf istype-le istype-false decidable__equal_int intformeq_wf itermMultiply_wf int_formula_prop_eq_lemma int_term_value_mul_lemma mul-commutes div-cancel nequal_wf equal_wf squash_wf true_wf istype-universe exp_add exp1 subtype_rel_self iff_weakening_equal rational-approx-property decidable__lt multiply-is-int-iff false_wf atom_subtype_base req_weakening req_wf rleq_wf rabs_wf rsub_wf int-rdiv_wf int-to-real_wf rdiv_wf rless-int rless_wf nat_plus_inc_int_nzero rleq_functionality rabs_functionality rsub_functionality int-rdiv-req rleq-int-fractions rleq_functionality_wrt_implies rleq_weakening_equal valueall-type-has-valueall product-valueall-type istype-atom set-valueall-type atom-valueall-type real-valueall-type int-valueall-type evalall-reduce
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut sqequalRule introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality because_Cache hypothesis dependent_set_memberEquality_alt independent_pairFormation imageMemberEquality hypothesisEquality baseClosed inhabitedIsType equalityIsType1 equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination universeIsType callbyvalueReduce independent_isectElimination intEquality lambdaEquality_alt closedConclusion applyEquality instantiate cumulativity rename setElimination equalityIsType4 baseApply unionElimination approximateComputation dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination multiplyEquality imageElimination universeEquality divideEquality productElimination applyLambdaEquality pointwiseFunctionality promote_hyp independent_pairEquality tokenEquality inrFormation_alt productEquality setEquality atomEquality setIsType

\mforall{}d:\mBbbN{}\msupplus{}.  \mforall{}x:\mBbbR{}.
    (d  decimal  digits  of  x    \mmember{}  \{a:Atom|  a  =  "display-as"\} 
      \mtimes{}  \{a:Atom|  a  =  "decimal-rational"\} 
      \mtimes{}  \{z:\mBbbR{}|  z  =  x\} 
      \mtimes{}  \{n:\mBbbN{}\msupplus{}|  n  =  d\} 
      \mtimes{}  \{n:\mBbbZ{}|  |x  -  (r(n)/r(10\^{}d))|  \mleq{}  (r(2)/r(10\^{}d))\}  )

Date html generated: 2019_10_30-AM-07_52_34
Last ObjectModification: 2018_11_08-PM-02_14_25

Theory : reals

Home Index