Nuprl Lemma : rneq-if-rabs2
∀x,y:ℝ. ((r0 < |x - y|) ⇒ x ≠ y)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
ge: i ≥ j ,
nat: ℕ,
rev_implies: P ⇐ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
subtype_rel: A ⊆r B,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
absval: |i|,
uiff: uiff(P;Q),
rneq: x ≠ y,
squash: ↓T,
less_than': less_than'(a;b),
less_than: a < b,
iff: P ⇐⇒ Q,
nequal: a ≠ b ∈ T ,
true: True,
int_nzero: ℤ-o,
and: P ∧ Q,
real: ℝ,
rminus: -(x),
rmax: rmax(x;y),
rsub: x - y,
has-value: (a)↓,
guard: {T},
sq_type: SQType(T),
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
int-to-real: r(n),
sq_exists: ∃x:A [B[x]],
rless: x < y,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
int_formula_prop_or_lemma,
int_formula_prop_le_lemma,
intformor_wf,
intformle_wf,
multiply-is-int-iff,
nat_properties,
absval_wf,
absval_unfold2,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
remainder_wfa,
rem_bounds_absval,
div_rem_sum,
radd-approx,
false_wf,
int_term_value_minus_lemma,
itermMinus_wf,
minus-is-int-iff,
add-is-int-iff,
less_than_wf,
istype-top,
nequal_wf,
istype-less_than,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformless_wf,
intformand_wf,
decidable__lt,
divide_wfa,
imax_strict_ub,
rminus_wf,
rabs-as-rmax,
int-value-type,
value-type-has-value,
real_wf,
rsub_wf,
rabs_wf,
int-to-real_wf,
rless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_not_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
itermAdd_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
nat_plus_properties,
int_subtype_base,
subtype_base_sq
Rules used in proof :
functionIsType,
inrFormation_alt,
inlFormation_alt,
equalityElimination,
unionIsType,
baseApply,
promote_hyp,
pointwiseFunctionality,
applyLambdaEquality,
inrEquality_alt,
inlEquality_alt,
imageElimination,
imageMemberEquality,
isectIsTypeImplies,
axiomSqEquality,
isect_memberFormation_alt,
lessCases,
productElimination,
sqequalBase,
baseClosed,
equalityIstype,
minusEquality,
independent_pairFormation,
closedConclusion,
dependent_set_memberEquality_alt,
applyEquality,
addEquality,
multiplyEquality,
callbyvalueReduce,
inhabitedIsType,
equalitySymmetry,
equalityTransitivity,
universeIsType,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
natural_numberEquality,
unionElimination,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
independent_isectElimination,
intEquality,
cumulativity,
isectElimination,
extract_by_obid,
introduction,
instantiate,
sqequalRule,
hypothesis,
cut,
rename,
thin,
setElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x,y:\mBbbR{}. ((r0 < |x - y|) {}\mRightarrow{} x \mneq{} y)
Date html generated:
2019_11_06-PM-00_27_38
Last ObjectModification:
2019_11_05-PM-02_11_50
Theory : reals
Home
Index