Nuprl Lemma : rneq-if-rabs
∀x,y:ℝ. x ≠ y supposing r0 < |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,
uimplies: b supposing a,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
rless: x < y,
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
int-to-real: r(n),
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
real: ℝ,
has-value: (a)↓,
rsub: x - y,
rmax: rmax(x;y),
rminus: -(x),
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
iff: P ⇐⇒ Q,
less_than': less_than'(a;b),
rneq: x ≠ y,
uiff: uiff(P;Q),
int_nzero: ℤ-o,
absval: |i|,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j
Lemmas referenced :
rlessw_wf,
int-to-real_wf,
rabs_wf,
rsub_wf,
rless_wf,
nat_plus_properties,
decidable__lt,
istype-less_than,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_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_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
real_wf,
value-type-has-value,
int-value-type,
rabs-as-rmax,
rminus_wf,
imax_strict_ub,
mul_nat_plus,
subtype_base_sq,
int_subtype_base,
istype-top,
less_than_wf,
add-is-int-iff,
minus-is-int-iff,
itermMinus_wf,
int_term_value_minus_lemma,
false_wf,
radd-approx,
div_rem_sum,
nequal_wf,
rem_bounds_absval,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
bool_wf,
assert-bnot,
iff_weakening_uiff,
assert_wf,
absval_unfold2,
absval_wf,
nat_properties,
multiply-is-int-iff,
intformle_wf,
intformor_wf,
int_formula_prop_le_lemma,
int_formula_prop_or_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
dependent_pairFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
dependent_set_memberEquality_alt,
hypothesisEquality,
universeIsType,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
because_Cache,
sqequalRule,
inhabitedIsType,
unionElimination,
imageElimination,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce,
intEquality,
multiplyEquality,
divideEquality,
addEquality,
minusEquality,
instantiate,
cumulativity,
equalityIsType4,
baseClosed,
lessCases,
axiomSqEquality,
isectIsTypeImplies,
imageMemberEquality,
inlEquality_alt,
inrEquality_alt,
applyLambdaEquality,
closedConclusion,
pointwiseFunctionality,
promote_hyp,
baseApply,
unionIsType,
remainderEquality,
equalityElimination,
inlFormation_alt,
inrFormation_alt,
functionIsType
Latex:
\mforall{}x,y:\mBbbR{}. x \mneq{} y supposing r0 < |x - y|
Date html generated:
2019_10_29-AM-09_39_21
Last ObjectModification:
2018_11_11-PM-11_11_25
Theory : reals
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