Nuprl Lemma : rat-cube-third-complex
∀k,n:ℕ. ∀K:n-dim-complex. ∀c:ℚCube(k).
((c ∈ K) ⇒ (∀p:ℝ^k. (in-rat-cube(k;p;c) ⇒ rat-cube-third(k;p;c) ⇒ (∀j:ℕ. (¬¬(∀d∈K'^(j).rat-cube-third(k;p;d)))))))
Proof
Definitions occuring in Statement :
rat-cube-third: rat-cube-third(k;p;c),
in-rat-cube: in-rat-cube(k;p;c),
real-vec: ℝ^n,
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
rational-cube-complex: n-dim-complex,
rational-cube: ℚCube(k)
Definitions unfolded in proof :
rat-cube-third: rat-cube-third(k;p;c),
true: True,
squash: ↓T,
cand: A c∧ B,
compatible-rat-cubes: Compatible(c;d),
int_seg: {i..j-},
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
stable: Stable{P},
subtype_rel: A ⊆r B,
decidable: Dec(P),
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
rational-cube-complex: n-dim-complex,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
lt_int: i <z j,
rat-complex-iter-subdiv: Error :rat-complex-iter-subdiv,
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rat-cube-third-half,
member-rat-complex-subdiv2,
rat-cube-third-not-in-face,
istype-universe,
equal_wf,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
rat-cube-face_wf,
rat-cube-face-dimension-equal,
decidable__equal_rc,
in-rat-cube-intersection,
rat-cube-intersection_wf,
inhabited-iff-in-rat-cube,
le_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
equal-wf-base,
compatible-rat-cubes-refl,
compatible-rat-cubes-symm,
compatible-rat-cubes_wf,
Error :pairwise-iff,
stable__false,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
l_all_wf2,
false_wf,
stable__not,
istype-nat,
rational-cube-complex_wf,
real-vec_wf,
in-rat-cube_wf,
istype-le,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
decidable__le,
subtract_wf,
Error :rat-complex-iter-subdiv_wf,
rat-complex-subdiv_wf,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
subtract-1-ge-0,
not_wf,
l_all_iff,
rational-cube_wf,
l_member_wf,
rat-cube-third_wf,
Error :not-not-l_all-shift,
primrec-unroll,
istype-less_than,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties
Rules used in proof :
applyLambdaEquality,
productIsType,
hyp_replacement,
universeEquality,
baseClosed,
imageMemberEquality,
imageElimination,
addEquality,
minusEquality,
intEquality,
unionIsType,
functionEquality,
unionEquality,
applyEquality,
dependent_set_memberEquality_alt,
cumulativity,
instantiate,
promote_hyp,
equalityIstype,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
functionIsType,
productElimination,
setIsType,
because_Cache,
inhabitedIsType,
functionIsTypeImplies,
universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k,n:\mBbbN{}. \mforall{}K:n-dim-complex. \mforall{}c:\mBbbQ{}Cube(k).
((c \mmember{} K)
{}\mRightarrow{} (\mforall{}p:\mBbbR{}\^{}k
(in-rat-cube(k;p;c)
{}\mRightarrow{} rat-cube-third(k;p;c)
{}\mRightarrow{} (\mforall{}j:\mBbbN{}. (\mneg{}\mneg{}(\mforall{}d\mmember{}K'\^{}(j).rat-cube-third(k;p;d)))))))
Date html generated:
2019_11_04-PM-04_43_37
Last ObjectModification:
2019_11_04-PM-04_17_24
Theory : real!vectors
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