Nuprl Lemma : rational-truncate1
∀q:ℚ. ∀e:{e:ℚ| 0 < e ∧ e < 1} . (∃p:ℤ × ℕ+ [((|q - (fst(p)/snd(p))| ≤ e) ∧ (((snd(p)) * e) ≤ 2))])
Proof
Definitions occuring in Statement :
qabs: |r|,
qle: r ≤ s,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
qmul: r * s,
rationals: ℚ,
nat_plus: ℕ+,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
set: {x:A| B[x]} ,
product: x:A × B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
has-value: (a)↓,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
guard: {T},
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
true: True,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
btrue: tt,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
less_than: a < b,
less_than': less_than'(a;b),
or: P ∨ Q,
sq_type: SQType(T)
Lemmas referenced :
value-type-has-value,
qless_wf,
subtract_wf,
qdiv_wf,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
equal-wf-T-base,
rationals_wf,
qle_wf,
int-subtype-rationals,
set-value-type,
int-value-type,
rat-int-bound_wf,
set_wf,
equal_wf,
less_than_wf,
qmul_wf,
squash_wf,
sq_stable__and,
sq_stable__less_than,
sq_stable_from_decidable,
decidable__qle,
qle_witness,
qmul_preserves_qless,
true_wf,
qmul_zero_qrng,
qmul-qdiv-cancel,
iff_weakening_equal,
qless-int,
qmul_preserves_qle2,
qle_weakening_lt_qorder,
qmul_com,
qadd_preserves_qle,
qsub-sub,
qsub_wf,
qless_witness,
qadd_wf,
qadd_preserves_qless,
qmul_comm_qrng,
qmul_one_qrng,
qadd_assoc,
mon_ident_q,
qmul_over_plus_qrng,
qmul_over_minus_qrng,
qadd_comm_q,
qadd_inv_assoc_q,
integer-part_wf,
subtype_rel_set,
qabs_wf,
int_nzero-rational,
subtype_rel_sets,
nequal_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
qabs-qdiv,
not_wf,
qmul-preserves-eq,
qabs-abs,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
intformnot_wf,
int_formula_prop_not_lemma,
integer-fractional-parts,
fractional-part_wf,
mon_assoc_q,
qadd_ac_1_q,
qinverse_q,
qabs-of-nonneg,
qmul_preserves_qle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
setElimination,
thin,
rename,
introduction,
sqequalRule,
callbyvalueReduce,
sqequalHypSubstitution,
productElimination,
cut,
extract_by_obid,
isectElimination,
setEquality,
intEquality,
productEquality,
hypothesisEquality,
natural_numberEquality,
hypothesis,
applyEquality,
because_Cache,
independent_isectElimination,
voidElimination,
baseClosed,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
independent_pairFormation,
independent_pairEquality,
imageMemberEquality,
imageElimination,
universeEquality,
isect_memberFormation,
minusEquality,
dependent_set_memberEquality,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
voidEquality,
computeAll,
hyp_replacement,
unionElimination,
equalityElimination,
lessCases,
sqequalAxiom,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}q:\mBbbQ{}. \mforall{}e:\{e:\mBbbQ{}| 0 < e \mwedge{} e < 1\} . (\mexists{}p:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{} [((|q - (fst(p)/snd(p))| \mleq{} e) \mwedge{} (((snd(p)) * e) \mleq{} 2))])
Date html generated:
2018_05_22-AM-00_31_11
Last ObjectModification:
2017_07_26-PM-06_58_52
Theory : rationals
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