Nuprl Lemma : not-proj-sep
∀n:ℕ. ∀a,b:ℙ^n. (¬a ≠ b ⇐⇒ req-vec(n + 1;u(a);u(b)) ∨ req-vec(n + 1;u(a);r(-1)*u(b)))
Proof
Definitions occuring in Statement :
proj-sep: a ≠ b,
punit: u(a),
real-proj: ℙ^n,
real-vec-mul: a*X,
req-vec: req-vec(n;x;y),
int-to-real: r(n),
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
real-vec-sep: a ≠ b,
real-vec-dist: d(x;y),
real-vec-mul: a*X,
real-vec-sub: X - Y,
req-vec: req-vec(n;x;y),
subtype_rel: A ⊆r B,
real-vec: ℝ^n,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
absval: |i|,
true: True,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
sq_stable: SqStable(P),
nat_plus: ℕ+,
stable: Stable{P},
proj-sep: a ≠ b
Lemmas referenced :
not_wf,
proj-sep_wf,
or_wf,
req-vec_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
punit_wf,
real-vec-mul_wf,
int-to-real_wf,
real-proj_wf,
nat_wf,
real-vec-sep-cases,
int_seg_wf,
rsub_wf,
rmul_wf,
real-vec_wf,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
itermSubtract_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
real-vec-norm_wf,
real-vec-sub_wf,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rless_functionality,
req_weakening,
real-vec-norm_functionality,
rabs_wf,
real-vec-norm-mul,
rless-int,
absval_wf,
sq_stable__less_than,
real_wf,
nat_plus_properties,
rless_wf,
rmul-int,
rmul_functionality,
rabs-int,
punit-norm1,
real-vec-dist_wf,
req_wf,
stable__req-vec,
false_wf,
real-vec-sep_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
real-vec-sep-symmetry,
not-real-vec-sep-iff-eq,
real-vec-sep_functionality,
req-vec_weakening,
not-real-vec-sep-refl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
sqequalRule,
because_Cache,
minusEquality,
applyEquality,
productElimination,
multiplyEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addLevel,
inlFormation,
setEquality,
equalityTransitivity,
equalitySymmetry,
inrFormation,
functionEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbP{}\^{}n. (\mneg{}a \mneq{} b \mLeftarrow{}{}\mRightarrow{} req-vec(n + 1;u(a);u(b)) \mvee{} req-vec(n + 1;u(a);r(-1)*u(b)))
Date html generated:
2017_10_05-AM-00_17_47
Last ObjectModification:
2017_07_28-AM-08_55_26
Theory : inner!product!spaces
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