Nuprl Lemma : rleq-real-vec-dist
∀[n:ℕ]. ∀[x,y:ℝ^n]. ∀[i:ℕn]. (|(x i) - y i| ≤ d(x;y))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec: ℝ^n,
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
real-vec: ℝ^n,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
real-vec-dist: d(x;y),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
uimplies: b supposing a,
nat: ℕ,
less_than': less_than'(a;b),
not: ¬A,
false: False,
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
dot-product: x⋅y,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
so_apply: x[s],
rsub: x - y,
radd: a + b,
accelerate: accelerate(k;f),
real-vec-sub: X - Y
Lemmas referenced :
square-rleq-implies,
rabs_wf,
rsub_wf,
real-vec-dist_wf,
real-vec-dist-nonneg,
le_witness_for_triv,
int_seg_wf,
real-vec_wf,
istype-nat,
rnexp_wf,
istype-void,
istype-le,
real-vec-norm_wf,
real-vec-sub_wf,
dot-product_wf,
rmul_wf,
rleq_functionality,
req_weakening,
real-vec-norm-squared,
rabs-rnexp2,
rnexp2,
item-rleq-rsum-of-nonneg,
subtract_wf,
subtype_rel_self,
real_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
istype-less_than,
square-nonneg,
subtract-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
applyEquality,
hypothesisEquality,
hypothesis,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
independent_functionElimination,
productElimination,
independent_isectElimination,
functionIsTypeImplies,
universeIsType,
natural_numberEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
dependent_set_memberEquality_alt,
independent_pairFormation,
lambdaFormation_alt,
voidElimination,
because_Cache,
imageElimination,
functionEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
productIsType,
addEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. \mforall{}[i:\mBbbN{}n]. (|(x i) - y i| \mleq{} d(x;y))
Date html generated:
2019_10_30-AM-08_30_28
Last ObjectModification:
2019_06_26-PM-06_55_23
Theory : reals
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