Nuprl Lemma : 0-dim-complex-polyhedron
∀k:ℕ. ∀K:0-dim-complex. ∀x:|K|. ∃i:ℕ||K||. req-vec(k;x;λj.rat2real(fst((K[i] j))))
Proof
Definitions occuring in Statement :
rat-cube-complex-polyhedron: |K|,
req-vec: req-vec(n;x;y),
rat2real: rat2real(q),
select: L[n],
length: ||as||,
int_seg: {i..j-},
nat: ℕ,
pi1: fst(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
lambda: λx.A[x],
natural_number: $n,
rational-cube-complex: n-dim-complex
Definitions unfolded in proof :
inject: Inj(A;B;f),
cons: [a / b],
nat_plus: ℕ+,
sq_stable: SqStable(P),
real: ℝ,
sq_exists: ∃x:A [B[x]],
rless: x < y,
real-vec-sep: a ≠ b,
sq_type: SQType(T),
guard: {T},
true: True,
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
l_member: (x ∈ l),
stable: Stable{P},
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
so_apply: x[s],
so_lambda: λ2x.t[x],
less_than': less_than'(a;b),
real-vec: ℝ^n,
pi2: snd(t),
pi1: fst(t),
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
l_all: (∀x∈L.P[x]),
in-rat-cube: in-rat-cube(k;p;c),
req-vec: req-vec(n;x;y),
prop: ℙ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
uimplies: b supposing a,
int_seg: {i..j-},
rational-cube-complex: n-dim-complex,
uall: ∀[x:A]. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
stable-union: Error :stable-union,
rat-cube-complex-polyhedron: |K|,
all: ∀x:A. B[x]
Lemmas referenced :
top_wf,
subtype_rel_list,
select-map,
map-length,
iff_weakening_equal,
istype-universe,
true_wf,
squash_wf,
rat-cube-dimension_wf,
equal-wf-base,
l_all_iff,
l_member_wf,
no_repeats_map,
sq_stable__no_repeats,
map_wf,
stable_req,
stable__all,
length_wf_nat,
add_nat_plus,
zero-add,
add-member-int_seg2,
select-cons-tl,
non_neg_length,
not-real-vec-sep-refl,
req-vec_weakening,
req-vec_inversion,
real-vec-sep_functionality,
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
add-is-int-iff,
nat_plus_properties,
real-vec-dist_wf,
int-to-real_wf,
sq_stable__less_than,
length_of_cons_lemma,
real-vec-sep-cases,
real-vec-sep_wf,
int_seg-case,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
int_subtype_base,
lelt_wf,
set_subtype_base,
subtype_base_sq,
le-add-cancel2,
add-commutes,
add-zero,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
add-associates,
condition-implies-le,
not-le-2,
int_seg_subtype,
subtype_rel_function,
int_term_value_subtract_lemma,
itermSubtract_wf,
decidable__equal_rationals,
primrec-wf2,
subtract_wf,
real-vec-sep-iff-rneq,
rneq-rat2real,
not_wf,
rneq_wf,
iff_weakening_uiff,
istype-less_than,
istype-false,
int_seg_subtype_nat,
cons_wf,
no_repeats_cons,
nil_wf,
istype-base,
stuck-spread,
length_of_nil_lemma,
list_wf,
exists_wf,
stable_wf,
no_repeats_wf,
rationals_wf,
list_induction,
istype-nat,
rational-cube-complex_wf,
rat-cube-complex-polyhedron_wf,
req-vec_wf,
req_wf,
rleq_antisymmetry,
equal_wf,
rat2real_wf,
rleq_wf,
rational-interval_wf,
subtype_rel_self,
rat-cube-dimension-zero,
int_formula_prop_less_lemma,
intformless_wf,
istype-le,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
select_wf,
in-rat-cube_wf,
rational-cube_wf,
length_wf,
int_seg_wf
Rules used in proof :
universeEquality,
baseApply,
pointwiseFunctionality,
imageMemberEquality,
intEquality,
cumulativity,
instantiate,
multiplyEquality,
minusEquality,
addEquality,
functionExtensionality,
productEquality,
setIsType,
functionIsTypeImplies,
isect_memberFormation_alt,
baseClosed,
closedConclusion,
equalityIstype,
independent_pairEquality,
applyLambdaEquality,
hyp_replacement,
promote_hyp,
inhabitedIsType,
functionEquality,
applyEquality,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality_alt,
independent_pairFormation,
isect_memberEquality_alt,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
dependent_functionElimination,
imageElimination,
productElimination,
independent_isectElimination,
because_Cache,
hypothesisEquality,
natural_numberEquality,
isectElimination,
extract_by_obid,
introduction,
universeIsType,
productIsType,
functionIsType,
sqequalRule,
voidElimination,
independent_functionElimination,
hypothesis,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}K:0-dim-complex. \mforall{}x:|K|. \mexists{}i:\mBbbN{}||K||. req-vec(k;x;\mlambda{}j.rat2real(fst((K[i] j))))
Date html generated:
2019_10_30-AM-10_13_19
Last ObjectModification:
2019_10_27-PM-02_51_55
Theory : real!vectors
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