Nuprl Lemma : qtruncate-property
∀[q:ℚ]. ∀[N:ℕ+]. (qtruncate(q;N) - (1/N) < q ∧ (q ≤ qtruncate(q;N)))
Proof
Definitions occuring in Statement :
qtruncate: qtruncate(q;N),
qle: r ≤ s,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
and: P ∧ Q,
int_nzero: ℤ-o,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
qtruncate: qtruncate(q;N),
cand: A c∧ B,
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q),
true: True,
qsub: r - s,
squash: ↓T,
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
qle_wf,
qmul_one_qrng,
iff_weakening_equal,
qmul-qdiv-cancel,
qadd_comm_q,
qmul_comm_qrng,
not_wf,
qmul_over_minus_qrng,
qadd_wf,
qmul_over_plus_qrng,
true_wf,
squash_wf,
qless_wf,
q-ceil_wf,
qmul_preserves_qle,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
qless-int,
qmul_preserves_qless,
nat_plus_wf,
qle_witness,
equal_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_plus_properties,
nequal_wf,
subtype_rel_sets,
int_nzero-rational,
qdiv_wf,
qtruncate_wf,
qsub_wf,
qless_witness,
int-subtype-rationals,
less_than_wf,
rationals_wf,
subtype_rel_set,
qmul_wf,
q-ceil-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
intEquality,
hypothesis,
lambdaEquality,
natural_numberEquality,
independent_isectElimination,
productElimination,
independent_pairEquality,
because_Cache,
setElimination,
rename,
setEquality,
lambdaFormation,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
unionElimination,
minusEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[q:\mBbbQ{}]. \mforall{}[N:\mBbbN{}\msupplus{}]. (qtruncate(q;N) - (1/N) < q \mwedge{} (q \mleq{} qtruncate(q;N)))
Date html generated:
2016_05_15-PM-11_35_27
Last ObjectModification:
2016_01_16-PM-09_12_40
Theory : rationals
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