Nuprl Lemma : not-all-nonneg-or-nonpos
¬(∀x:ℝ. ((r0 ≤ x) ∨ (x ≤ r0)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
or: P ∨ Q,
isl: isl(x),
and: P ∧ Q,
cand: A c∧ B,
false: False,
uall: ∀[x:A]. B[x],
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
nat: ℕ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_exists: ∃x:A [B[x]],
converges-to: lim n→∞.x[n] = y,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
sq_stable: SqStable(P),
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
le: A ≤ B,
less_than': less_than'(a;b),
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
rtermMinus: rtermMinus(num),
rtermSubtract: left "-" right,
pi2: snd(t),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
assert: ↑b,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T)
Lemmas referenced :
real_wf,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
bool_wf,
rleq_wf,
int-to-real_wf,
better-continuity-for-reals,
rdiv_wf,
rless-int,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
rless_wf,
istype-nat,
nat_plus_subtype_nat,
istype-le,
rabs_wf,
rsub_wf,
nat_plus_properties,
nat_plus_wf,
sq_stable__rless,
rminus_wf,
rmul_preserves_rleq,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
rleq-int,
istype-false,
assert-rat-term-eq2,
rtermMinus_wf,
rtermSubtract_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
rleq-int-fractions,
istype-less_than,
decidable__le,
int_term_value_mul_lemma,
req_functionality,
rabs-of-nonpos,
req_weakening,
rleq_functionality,
req_transitivity,
rminus_functionality,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
iff_imp_equal_bool,
istype-assert,
subtype_base_sq,
bool_subtype_base,
istype-true,
rmul_preserves_rleq2,
rabs-of-nonneg,
assert_elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
rename,
dependent_pairFormation_alt,
lambdaEquality_alt,
applyEquality,
functionExtensionality,
sqequalHypSubstitution,
hypothesisEquality,
introduction,
extract_by_obid,
hypothesis,
inhabitedIsType,
thin,
unionElimination,
sqequalRule,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
because_Cache,
baseClosed,
sqequalBase,
independent_pairFormation,
voidElimination,
functionIsType,
productIsType,
isectElimination,
natural_numberEquality,
productElimination,
unionIsType,
setElimination,
equalityElimination,
closedConclusion,
minusEquality,
addEquality,
independent_isectElimination,
inrFormation_alt,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
dependent_set_memberFormation_alt,
dependent_set_memberEquality_alt,
imageMemberEquality,
imageElimination,
multiplyEquality,
instantiate,
cumulativity
Latex:
\mneg{}(\mforall{}x:\mBbbR{}. ((r0 \mleq{} x) \mvee{} (x \mleq{} r0)))
Date html generated:
2019_10_30-AM-07_19_44
Last ObjectModification:
2019_05_08-PM-07_14_45
Theory : reals
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