Nuprl Lemma : rsub-rmin-rleq-rabs
∀[a,b:ℝ]. ((b - rmin(a;b)) ≤ |a - b|)
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
rmin: rmin(x;y),
rsub: x - y,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
rsub: x - y,
rabs: |x|,
rmin: rmin(x;y),
rminus: -(x),
radd: a + b,
reg-seq-list-add: reg-seq-list-add(L),
accelerate: accelerate(k;f),
uimplies: b supposing a,
nat_plus: ℕ+,
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
nil: [],
it: ⋅,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
has-value: (a)↓,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
nat: ℕ,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
int_lower: {...i},
ge: i ≥ j
Lemmas referenced :
rleq-iff4,
rsub_wf,
rmin_wf,
rabs_wf,
nat_plus_wf,
less_than'_wf,
real_wf,
value-type-has-value,
int-value-type,
mul_nat_plus,
less_than_wf,
imin_wf,
equal_wf,
ifthenelse_wf,
le_int_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
absval_wf,
nat_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
le_wf,
squash_wf,
add_functionality_wrt_eq,
minus_functionality_wrt_eq,
imin_unfold,
iff_weakening_equal,
nat_plus_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_formula_prop_wf,
decidable__le,
div_2_to_1,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
div_bounds_1,
divide_wf,
absval-minus,
absval_pos,
div_bounds_2,
le_weakening2,
decidable__lt,
nat_properties,
intformless_wf,
int_formula_prop_less_lemma,
add_nat_wf,
false_wf,
add-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
lambdaFormation,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
callbyvalueReduce,
sqleReflexivity,
intEquality,
independent_isectElimination,
multiplyEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
addEquality,
addLevel,
instantiate,
cumulativity,
divideEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
imageElimination,
universeEquality,
int_eqEquality,
voidEquality,
computeAll,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[a,b:\mBbbR{}]. ((b - rmin(a;b)) \mleq{} |a - b|)
Date html generated:
2017_10_03-AM-08_26_50
Last ObjectModification:
2017_07_28-AM-07_24_35
Theory : reals
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