Nuprl Lemma : egyptian-fraction
∀r:{r:ℚ| (0 ≤ r) ∧ r < 1} . (∃L:ℕ+ List [(r = Σ0 ≤ i < ||L||. (1/L[i]) ∈ ℚ)])
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
qless: r < s,
qdiv: (r/s),
rationals: ℚ,
select: L[n],
length: ||as||,
list: T List,
nat_plus: ℕ+,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
le: A ≤ B,
uimplies: b supposing a,
guard: {T},
int_seg: {i..j-},
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
nat_plus: ℕ+,
so_apply: x[s],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_stable: SqStable(P),
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_type: SQType(T),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
less_than': less_than'(a;b),
subtract: n - m,
cand: A c∧ B
Lemmas referenced :
fractional-part-rep,
qle_wf,
qless_wf,
sq_exists_wf,
list_wf,
nat_plus_wf,
equal_wf,
rationals_wf,
qsum_wf,
length_wf,
qdiv_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,
subtype_rel_set,
less_than_wf,
int-subtype-rationals,
int_nzero-rational,
subtype_rel_sets,
nequal_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
int_seg_wf,
set_wf,
squash_wf,
sq_stable__and,
sq_stable_from_decidable,
decidable__qle,
decidable__qless,
qless_witness,
all_wf,
nat_wf,
le_wf,
lelt_wf,
natrec_wf,
decidable__equal_int,
subtype_base_sq,
nil_wf,
length_of_nil_lemma,
true_wf,
sum_unroll_base_q,
iff_weakening_equal,
qmul-preserves-eq,
qmul_wf,
qmul_zero_qrng,
qmul-qdiv-cancel,
div_rem_sum,
rem_bounds_1,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
zero-add,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
divide_wf,
less_than_transitivity2,
add-is-int-iff,
multiply-is-int-iff,
itermMultiply_wf,
itermAdd_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
add_nat_wf,
mul_preserves_lt,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
append_wf,
cons_wf,
length-append,
length_of_cons_lemma,
set_subtype_base,
length_append,
subtype_rel_list,
top_wf,
length-singleton,
non_neg_length,
sum_unroll_hi_q,
add_nat_plus,
length_wf_nat,
nat_plus_subtype_nat,
int_nzero_wf,
add-subtract-cancel,
select_append_back,
zero-le-nat,
select-cons-hd,
not_wf,
equal-wf-T-base,
select_append_front,
qadd_wf,
qmul-mul,
qadd-add,
int-equal-in-rationals,
qmul_over_plus_qrng,
qmul_assoc_qrng,
qmul_comm_qrng,
qmul_ac_1_qrng,
qmul-qdiv-cancel3,
qmul-qdiv-cancel2,
qmul_one_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
productElimination,
dependent_set_memberEquality,
hypothesisEquality,
independent_pairFormation,
hypothesis,
productEquality,
isectElimination,
natural_numberEquality,
applyEquality,
because_Cache,
sqequalRule,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
lambdaEquality,
independent_isectElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
setEquality,
equalityTransitivity,
baseClosed,
independent_functionElimination,
imageMemberEquality,
functionEquality,
functionExtensionality,
instantiate,
cumulativity,
comment,
dependent_set_memberFormation,
universeEquality,
addEquality,
minusEquality,
multiplyEquality,
divideEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
addLevel,
impliesFunctionality
Latex:
\mforall{}r:\{r:\mBbbQ{}| (0 \mleq{} r) \mwedge{} r < 1\} . (\mexists{}L:\mBbbN{}\msupplus{} List [(r = \mSigma{}0 \mleq{} i < ||L||. (1/L[i]))])
Date html generated:
2018_05_22-AM-00_32_45
Last ObjectModification:
2017_07_26-PM-06_59_18
Theory : rationals
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