Nuprl Lemma : real-vec-dist-monotone-in-dim
∀[m:ℕ+]. ∀[p,q:ℝ^m]. (d(p;q) ≤ d(p;q))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec: ℝ^n,
rleq: x ≤ y,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
real-vec-dist: d(x;y),
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
dot-product: x⋅y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
so_lambda: λ2x.t[x],
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
req_int_terms: t1 ≡ t2
Lemmas referenced :
square-rleq-implies,
real-vec-dist_wf,
subtract_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
real-vec-subtype,
nat_plus_subtype_nat,
real-vec-dist-nonneg,
rleq_functionality,
rnexp_wf,
real-vec-norm_wf,
real-vec-sub_wf,
dot-product_wf,
real-vec-norm-squared,
le_witness_for_triv,
real-vec_wf,
nat_plus_wf,
rsum-split-last,
rsum_wf,
rmul_wf,
int_seg_properties,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
istype-less_than,
int_seg_wf,
radd_wf,
subtract-add-cancel,
radd-preserves-rleq,
rminus_wf,
int-to-real_wf,
itermMinus_wf,
itermMultiply_wf,
square-nonneg,
req_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
dependent_set_memberEquality_alt,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
hypothesisEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
applyEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
productElimination,
equalityIstype,
functionIsTypeImplies,
isectIsTypeImplies,
imageElimination,
productIsType,
addEquality,
closedConclusion
Latex:
\mforall{}[m:\mBbbN{}\msupplus{}]. \mforall{}[p,q:\mBbbR{}\^{}m]. (d(p;q) \mleq{} d(p;q))
Date html generated:
2019_10_30-AM-08_30_55
Last ObjectModification:
2019_06_28-PM-03_24_15
Theory : reals
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