Nuprl Lemma : mdist-max-metric-mul2

∀[n:ℕ]. ∀[p,q:ℝ^n]. ∀[c:ℝ].  (mdist(max-metric(n);c*p;c*q) = (|c| * mdist(max-metric(n);p;q)))


Proof



Definitions occuring in Statement :  max-metric: max-metric(n),  real-vec-mul: a*X,  real-vec: ℝ^n,  mdist: mdist(d;x;y),  rabs: |x|,  req: x = y,  rmul: a * b,  real: ℝ,  nat: ℕ,  uall: ∀[x:A]. B[x]
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,  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,  and: P ∧ Q,  prop: ℙ,  le: A ≤ B,  less_than': less_than'(a;b),  decidable: Dec(P),  or: P ∨ Q,  squash: ↓T,  uiff: uiff(P;Q),  subtype_rel: A ⊆r B,  int_seg: {i..j-},  lelt: i ≤ j < k,  req_int_terms: t1 ≡ t2,  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  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,  rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :  sq_stable__req,  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,  req_witness,  primrec0_lemma,  real-vec_wf,  istype-le,  subtract-1-ge-0,  decidable__le,  intformnot_wf,  int_formula_prop_not_lemma,  istype-nat,  itermSubtract_wf,  itermMultiply_wf,  req-iff-rsub-is-0,  lt_int_wf,  real-vec-subtype,  subtract_wf,  int_term_value_subtract_lemma,  decidable__lt,  req_wf,  real_polynomial_null,  real_term_value_sub_lemma,  real_term_value_const_lemma,  real_term_value_mul_lemma,  real_term_value_var_lemma,  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,  zero-rleq-rabs,  rmax_functionality,  radd_wf,  rminus_wf,  itermAdd_wf,  itermMinus_wf,  req_weakening,  req_functionality,  rmul-rmax,  req_inversion,  rabs-rmul,  rabs_functionality,  real_term_value_add_lemma,  real_term_value_minus_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,  inhabitedIsType,  universeIsType,  setElimination,  rename,  because_Cache,  independent_functionElimination,  intWeakElimination,  lambdaFormation_alt,  independent_isectElimination,  approximateComputation,  dependent_pairFormation_alt,  int_eqEquality,  dependent_functionElimination,  isect_memberEquality_alt,  voidElimination,  independent_pairFormation,  functionIsTypeImplies,  dependent_set_memberEquality_alt,  unionElimination,  imageMemberEquality,  baseClosed,  imageElimination,  productElimination,  closedConclusion,  productIsType,  equalityIstype,  equalityTransitivity,  equalitySymmetry,  equalityElimination,  promote_hyp,  instantiate,  cumulativity
Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[p,q:\mBbbR{}\^{}n].  \mforall{}[c:\mBbbR{}].    (mdist(max-metric(n);c*p;c*q)  =  (|c|  *  mdist(max-metric(n);p;q)))


Date html generated: 2019_10_30-AM-08_39_38
Last ObjectModification: 2019_10_02-AM-11_04_29

Theory : reals


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