Nuprl Lemma : egyptian-number
∀q:ℚ. (∃p:{ℤ × (ℕ+ List)| let x,L = p in q = (x + Σ0 ≤ i < ||L||. (1/L[i])) ∈ ℚ})
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qdiv: (r/s),
qadd: 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]},
spread: spread def,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
sq_exists: ∃x:{A| B[x]},
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
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
Lemmas referenced :
integer-fractional-parts,
egyptian-fraction,
fractional-part_wf,
integer-part_wf,
equal_wf,
rationals_wf,
qadd_wf,
int-subtype-rationals,
qsum_wf,
length_wf,
nat_plus_wf,
qdiv_wf,
select_wf,
int_seg_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_nzero-rational,
subtype_rel_sets,
nequal_wf,
nat_plus_properties,
set_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
int_seg_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
dependent_set_memberFormation,
independent_pairEquality,
productElimination,
sqequalRule,
isectElimination,
applyEquality,
natural_numberEquality,
lambdaEquality,
because_Cache,
independent_isectElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
setEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_functionElimination,
hyp_replacement,
dependent_set_memberEquality
Latex:
\mforall{}q:\mBbbQ{}. (\mexists{}p:\{\mBbbZ{} \mtimes{} (\mBbbN{}\msupplus{} List)| let x,L = p in q = (x + \mSigma{}0 \mleq{} i < ||L||. (1/L[i]))\})
Date html generated:
2016_10_26-AM-06_48_28
Last ObjectModification:
2016_07_12-AM-08_01_58
Theory : rationals
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