Nuprl Lemma : num-digits_wf
∀[k:ℕ]. (num-digits(k) ∈ {n:ℕ+| ((10^n - 1 ≤ k) ∨ (k = 0 ∈ ℤ)) ∧ k < 10^n} )
Proof
Definitions occuring in Statement :
num-digits: num-digits(k),
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
num-digits: num-digits(k),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
true: True,
squash: ↓T,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nat_plus: ℕ+,
cand: A c∧ B,
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
has-value: (a)↓,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_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,
ge_wf,
less_than_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
exp0_lemma,
decidable__or,
equal-wf-base,
int_subtype_base,
intformor_wf,
int_formula_prop_or_lemma,
squash_wf,
true_wf,
exp1,
iff_weakening_equal,
or_wf,
exp_wf2,
nat_plus_properties,
equal-wf-T-base,
nat_plus_subtype_nat,
value-type-has-value,
int-value-type,
div_rem_sum,
nequal_wf,
rem_bounds_1,
add-is-int-iff,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
set_wf,
nat_plus_wf,
not-lt-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-zero,
add-commutes,
zero-add,
le-add-cancel,
set_subtype_base,
exp-positive,
exp_add,
exp_step
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productElimination,
unionElimination,
applyEquality,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
equalityElimination,
lessCases,
sqequalAxiom,
imageMemberEquality,
baseClosed,
imageElimination,
promote_hyp,
instantiate,
cumulativity,
addEquality,
universeEquality,
productEquality,
callbyvalueReduce,
divideEquality,
addLevel,
pointwiseFunctionality,
baseApply,
closedConclusion,
multiplyEquality,
inlFormation
Latex:
\mforall{}[k:\mBbbN{}]. (num-digits(k) \mmember{} \{n:\mBbbN{}\msupplus{}| ((10\^{}n - 1 \mleq{} k) \mvee{} (k = 0)) \mwedge{} k < 10\^{}n\} )
Date html generated:
2017_10_04-PM-11_01_34
Last ObjectModification:
2017_06_02-PM-00_14_37
Theory : reals_2
Home
Index