Nuprl Lemma : ratreal-negative
∀r:ℤ × ℕ+. (ratreal(r) < r0 ⇐⇒ fst(r) < 0)
Proof
Definitions occuring in Statement :
ratreal: ratreal(r),
rless: x < y,
int-to-real: r(n),
nat_plus: ℕ+,
less_than: a < b,
pi1: fst(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
product: x:A × B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
pi1: fst(t),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2
Lemmas referenced :
istype-int,
nat_plus_wf,
ratreal_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
istype-less_than,
rless_functionality,
ratreal-req,
req_weakening,
rless-int-fractions3,
itermMultiply_wf,
int_term_value_mul_lemma,
rmul_preserves_rless,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
req_transitivity,
rmul_functionality,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
productElimination,
thin,
sqequalRule,
productIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
universeIsType,
sqequalHypSubstitution,
isectElimination,
independent_pairEquality,
hypothesisEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
promote_hyp,
multiplyEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. (ratreal(r) < r0 \mLeftarrow{}{}\mRightarrow{} fst(r) < 0)
Date html generated:
2019_10_30-AM-09_34_25
Last ObjectModification:
2019_01_13-PM-01_42_04
Theory : reals
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