Nuprl Lemma : in-0-dim-cube
∀[k:ℕ]. ∀[c:ℚCube(k)]. ∀[p:ℝ^k]. uiff(in-rat-cube(k;p;c);req-vec(k;p;λj.rat2real(fst((c j))))) supposing dim(c) = 0 ∈ ℤ
Proof
Definitions occuring in Statement :
in-rat-cube: in-rat-cube(k;p;c),
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rat2real: rat2real(q),
nat: ℕ,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
apply: f a,
lambda: λx.A[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
lelt: i ≤ j < k,
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
pi2: snd(t),
int_seg: {i..j-},
subtype_rel: A ⊆r B,
squash: ↓T,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
in-rat-cube: in-rat-cube(k;p;c),
req-vec: req-vec(n;x;y),
and: P ∧ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
so_apply: x[s],
nat: ℕ,
pi1: fst(t),
rational-interval: ℚInterval,
implies: P ⇒ Q,
all: ∀x:A. B[x],
rational-cube: ℚCube(k),
real-vec: ℝ^n,
so_lambda: λ2x.t[x],
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Lemmas referenced :
req_fake_le_antisymmetry,
real_term_value_const_lemma,
real_term_value_var_lemma,
istype-void,
real_term_value_sub_lemma,
int-to-real_wf,
real_polynomial_null,
iff_weakening_equal,
subtype_rel_self,
rationals_wf,
real_wf,
true_wf,
squash_wf,
req-iff-rsub-is-0,
itermVar_wf,
itermSubtract_wf,
rleq_weakening,
req_weakening,
rleq_functionality,
rleq_wf,
rat-cube-dimension-zero,
sq_stable__req,
req_wf,
sq_stable__all,
istype-nat,
rational-cube_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
istype-int,
le_witness_for_triv,
req_witness,
sq_stable__in-rat-cube,
sq_stable__uiff,
int_seg_wf,
rat2real_wf,
req-vec_wf,
in-rat-cube_wf,
uiff_wf,
real-vec_wf,
sq_stable__uall
Rules used in proof :
voidElimination,
int_eqEquality,
approximateComputation,
promote_hyp,
universeEquality,
instantiate,
productIsType,
functionIsType,
independent_pairFormation,
sqequalBase,
addEquality,
minusEquality,
intEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_isectElimination,
isectIsTypeImplies,
functionIsTypeImplies,
isect_memberEquality_alt,
independent_pairEquality,
rename,
setElimination,
natural_numberEquality,
universeIsType,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
productElimination,
lambdaFormation_alt,
inhabitedIsType,
applyEquality,
because_Cache,
lambdaEquality_alt,
sqequalRule,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)].
\mforall{}[p:\mBbbR{}\^{}k]. uiff(in-rat-cube(k;p;c);req-vec(k;p;\mlambda{}j.rat2real(fst((c j))))) supposing dim(c) = 0
Date html generated:
2019_10_30-AM-10_12_53
Last ObjectModification:
2019_10_29-PM-01_47_57
Theory : real!vectors
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