Nuprl Lemma : q_trichotomy
∀r:ℚ. ((↑qpositive(r)) ∨ (r = 0 ∈ ℚ) ∨ (↑qpositive(-(r))))
Proof
Definitions occuring in Statement :
qpositive: qpositive(r),
qmul: r * s,
rationals: ℚ,
assert: ↑b,
all: ∀x:A. B[x],
or: P ∨ Q,
minus: -n,
natural_number: $n,
equal: s = t ∈ T
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,
qdiv: (r/s),
top: Top,
ifthenelse: if b then t else f fi ,
btrue: tt,
mk-rational: mk-rational(a;b),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
bfalse: ff,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
q-elim,
nat_plus_properties,
assert-qeq,
int-subtype-rationals,
assert_wf,
qeq_wf2,
not_wf,
equal-wf-base,
rationals_wf,
int_subtype_base,
qinv-elim,
qmul-elim,
isint-int,
mk-rational_wf,
full-omega-unsat,
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,
qpositive-elim,
qeq-elim,
decidable__or,
or_wf,
less_than_wf,
decidable__cand,
decidable__lt,
decidable__equal_int,
intformnot_wf,
intformor_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_formula_prop_or_lemma,
int_term_value_mul_lemma,
iff_transitivity,
bor_wf,
band_wf,
lt_int_wf,
iff_weakening_uiff,
assert_of_bor,
assert_of_band,
assert_of_lt_int,
assert_of_eq_int,
eq_int_wf,
qpositive_wf,
qmul_wf,
equal-wf-T-base
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,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
multiplyEquality,
minusEquality,
isintReduceTrue,
productEquality,
baseApply,
closedConclusion,
unionElimination,
orFunctionality,
equalityTransitivity,
equalitySymmetry,
orLevelFunctionality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}r:\mBbbQ{}. ((\muparrow{}qpositive(r)) \mvee{} (r = 0) \mvee{} (\muparrow{}qpositive(-(r))))
Date html generated:
2018_05_21-PM-11_49_33
Last ObjectModification:
2017_07_26-PM-06_43_41
Theory : rationals
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