Nuprl Lemma : q-archimedean
∀a:ℚ. ∃n:ℕ. ((-(n) ≤ a) ∧ (a ≤ n))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qmul: r * s,
rationals: ℚ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
not: ¬A,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
le: A ≤ B,
int_lower: {...i},
gt: i > j,
ge: i ≥ j ,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
less_than: a < b,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
better-q-elim,
nat_plus_properties,
assert-qeq,
int-subtype-rationals,
assert_wf,
qeq_wf2,
not_wf,
equal-wf-base,
rationals_wf,
int_subtype_base,
div_rem_sum,
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,
nequal_wf,
qless-int,
decidable__le,
rem_bounds_1,
le_wf,
less_than_wf,
div_bounds_1,
rem_bounds_2,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
div_bounds_2,
exists_wf,
nat_wf,
qle_wf,
qmul_wf,
subtype_rel_set,
itermAdd_wf,
int_term_value_add_lemma,
qmul_preserves_qle,
qdiv_wf,
qmul-mul,
qle-int,
itermMultiply_wf,
int_term_value_mul_lemma,
squash_wf,
true_wf,
qmul_over_minus_qrng,
qmul_comm_qrng,
qmul-qdiv-cancel,
iff_weakening_equal,
itermMinus_wf,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
hypothesis,
setElimination,
rename,
addLevel,
impliesFunctionality,
applyEquality,
sqequalRule,
natural_numberEquality,
independent_isectElimination,
because_Cache,
baseClosed,
dependent_set_memberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
minusEquality,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
productEquality,
addEquality,
equalityTransitivity,
multiplyEquality,
imageElimination,
imageMemberEquality,
universeEquality,
independent_functionElimination
Latex:
\mforall{}a:\mBbbQ{}. \mexists{}n:\mBbbN{}. ((-(n) \mleq{} a) \mwedge{} (a \mleq{} n))
Date html generated:
2016_10_26-AM-06_35_16
Last ObjectModification:
2016_07_12-AM-07_58_10
Theory : rationals
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