### Nuprl Lemma : dd_wf

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

Proof

Definitions occuring in Statement :  dd: `n decimal digits of x ` rdiv: `(x/y)` rleq: `x ≤ y` rabs: `|x|` rsub: `x - y` req: `x = y` int-to-real: `r(n)` real: `ℝ` fastexp: `i^n` nat_plus: `ℕ+` all: `∀x:A. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` product: `x:A × B[x]` natural_number: `\$n` int: `ℤ` token: `"\$token"` atom: `Atom` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` dd: `n decimal digits of x ` uall: `∀[x:A]. B[x]` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` and: `P ∧ Q` implies: `P `` Q` has-value: `(a)↓` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` sq_type: `SQType(T)` guard: `{T}` exp: `i^n` primrec: `primrec(n;b;c)` subtract: `n - m` nat: `ℕ` 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: `ℙ` le: `A ≤ B` nequal: `a ≠ b ∈ T ` int_nzero: `ℤ-o` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` rational-approx: `(x within 1/n)` real: `ℝ` 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

Latex:
\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

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