Nuprl Lemma : mdist-max-metric-mul-rleq
∀[n:ℕ]. ∀[x:ℝ^n]. ∀[c,c':ℝ]. (mdist(max-metric(n);c*x;c'*x) ≤ (|c - c'| * mdist(max-metric(n);x;λi.r0)))
Proof
Definitions occuring in Statement :
max-metric: max-metric(n),
real-vec-mul: a*X,
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
max-metric: max-metric(n),
mdist: mdist(d;x;y),
real-vec-mul: a*X,
member: t ∈ T,
real-vec: ℝ^n,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
less_than: a < b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2
Lemmas referenced :
sq_stable__rleq,
primrec_wf,
real_wf,
int-to-real_wf,
rmax_wf,
rabs_wf,
rsub_wf,
rmul_wf,
int_seg_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
le_witness_for_triv,
primrec0_lemma,
rmul-nonneg-case1,
zero-rleq-rabs,
rleq_weakening_equal,
real-vec_wf,
istype-le,
subtract-1-ge-0,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-nat,
lt_int_wf,
real-vec-subtype,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
int_seg_properties,
rleq_wf,
primrec-unroll,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
rmax_lb,
rmax_ub,
rleq_functionality,
req_weakening,
rmul-rmax,
req_inversion,
rabs-rmul,
radd_wf,
rminus_wf,
itermMultiply_wf,
itermAdd_wf,
itermMinus_wf,
rabs_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
natural_numberEquality,
lambdaEquality_alt,
applyEquality,
because_Cache,
inhabitedIsType,
universeIsType,
setElimination,
rename,
productElimination,
independent_functionElimination,
intWeakElimination,
lambdaFormation_alt,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
dependent_set_memberEquality_alt,
unionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
closedConclusion,
productIsType,
equalityIstype,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbR{}\^{}n]. \mforall{}[c,c':\mBbbR{}].
(mdist(max-metric(n);c*x;c'*x) \mleq{} (|c - c'| * mdist(max-metric(n);x;\mlambda{}i.r0)))
Date html generated:
2019_10_30-AM-08_40_05
Last ObjectModification:
2019_10_02-AM-11_04_48
Theory : reals
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