Nuprl Lemma : egyptian_wf
∀q:ℚ. (egyptian(q) ∈ {p:ℤ × (ℕ+ List)| let x,L = p in q = (x + Σ0 ≤ i < ||L||. (1/L[i])) ∈ ℚ} )
Proof
Definitions occuring in Statement :
egyptian: egyptian(q),
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],
member: t ∈ T,
set: {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,
egyptian: egyptian(q),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
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,
prop: ℙ,
less_than: a < b,
squash: ↓T,
nat_plus: ℕ+,
so_apply: x[s],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_exists: ∃x:A [B[x]],
norm-pair: norm-pair(Na;Nb),
has-value: (a)↓,
sq_type: SQType(T),
id-fun: id-fun(T)
Lemmas referenced :
egyptian-number,
all_wf,
sq_exists_wf,
equal_wf,
qadd_wf,
qsum_wf,
qdiv_wf,
select_wf,
int_seg_properties,
length_wf,
nat_plus_wf,
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,
rationals_wf,
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,
value-type-has-value,
int-value-type,
subtype_base_sq,
list_wf,
list_subtype_base,
set_subtype_base,
norm-list_wf,
set-value-type,
id-fun_wf,
set_wf,
list-value-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
hypothesisEquality,
introduction,
isectElimination,
because_Cache,
productElimination,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
setEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
baseClosed,
independent_functionElimination,
callbyvalueReduce,
cumulativity,
dependent_set_memberEquality,
functionExtensionality,
independent_pairEquality
Latex:
\mforall{}q:\mBbbQ{}. (egyptian(q) \mmember{} \{p:\mBbbZ{} \mtimes{} (\mBbbN{}\msupplus{} List)| let x,L = p in q = (x + \mSigma{}0 \mleq{} i < ||L||. (1/L[i]))\} )
Date html generated:
2018_05_22-AM-00_32_54
Last ObjectModification:
2017_07_26-PM-06_59_31
Theory : rationals
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