Nuprl Lemma : rat-zero-cases
∀x:ℤ × ℕ+. ((↓r0 ≤ ratreal(x)) ∨ (↓ratreal(x) ≤ r0))
Proof
Definitions occuring in Statement :
ratreal: ratreal(r),
rleq: x ≤ y,
int-to-real: r(n),
nat_plus: ℕ+,
all: ∀x:A. B[x],
squash: ↓T,
or: P ∨ Q,
product: x:A × B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
prop: ℙ,
nat_plus: ℕ+,
decidable: Dec(P),
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
rat-rleq-cases-ext,
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
le: A ≤ B
Lemmas referenced :
istype-int,
nat_plus_wf,
rat-rleq-cases-ext,
subtype_rel_self,
squash_wf,
rleq_wf,
ratreal_wf,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
nat_plus_properties,
intformand_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int-to-real_wf,
rdiv_wf,
rless-int,
rless_wf,
rleq-int-fractions2,
istype-false,
rleq_transitivity,
rleq-int-fractions3,
rleq_functionality,
ratreal-req,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
productIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
universeIsType,
applyEquality,
thin,
instantiate,
sqequalRule,
sqequalHypSubstitution,
isectElimination,
functionEquality,
productEquality,
intEquality,
unionEquality,
hypothesisEquality,
independent_pairEquality,
natural_numberEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
productElimination,
multiplyEquality,
setElimination,
rename,
inhabitedIsType,
equalityElimination,
because_Cache,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isectIsTypeImplies,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
cumulativity,
int_eqEquality,
inlFormation_alt,
inrFormation_alt,
closedConclusion
Latex:
\mforall{}x:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. ((\mdownarrow{}r0 \mleq{} ratreal(x)) \mvee{} (\mdownarrow{}ratreal(x) \mleq{} r0))
Date html generated:
2019_10_30-AM-09_29_04
Last ObjectModification:
2019_01_11-AM-11_56_49
Theory : reals
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