Nuprl Lemma : meq-max-metric-iff-meq-rn-metric
∀[n:ℕ]. ∀[x,y:ℝ^n]. uiff(x ≡ y;x ≡ y)
Proof
Definitions occuring in Statement :
max-metric: max-metric(n),
rn-metric: rn-metric(n),
real-vec: ℝ^n,
meq: x ≡ y,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
meq: x ≡ y,
mdist: mdist(d;x;y),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
metric-leq: d1 ≤ d2,
all: ∀x:A. B[x],
nat: ℕ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
scale-metric: c*d,
req_int_terms: t1 ≡ t2
Lemmas referenced :
rn-metric-leq-max-metric,
max-metric-leq-rn-metric,
rleq_antisymmetry,
mdist_wf,
real-vec_wf,
rn-metric_wf,
int-to-real_wf,
mdist-nonneg,
req_witness,
req_wf,
max-metric_wf,
istype-nat,
scale-metric_wf,
rleq-int,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_var_lemma,
int_formula_prop_wf,
rleq_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rmul_wf,
rleq_weakening,
itermSubtract_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
rleq_functionality,
rmul_functionality,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
independent_pairFormation,
hypothesis,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
universeIsType,
dependent_functionElimination,
setElimination,
rename,
productElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. uiff(x \mequiv{} y;x \mequiv{} y)
Date html generated:
2019_10_30-AM-08_40_59
Last ObjectModification:
2019_10_02-AM-11_05_25
Theory : reals
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