Nuprl Lemma : positive-rat-cube-dimension
∀k:ℕ. ∀c:ℚCube(k). (0 < dim(c) ⇒ (∃i:ℕk. (dim(c i) = 1 ∈ ℤ)))
Proof
Definitions occuring in Statement :
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k),
rat-interval-dimension: dim(I),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rat-cube-dimension: dim(c),
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
nat: ℕ,
le: A ≤ B,
not: ¬A,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
rational-cube: ℚCube(k),
so_apply: x[s],
nat_plus: ℕ+,
true: True,
iff: P ⇐⇒ Q,
lelt: i ≤ j < k,
cand: A c∧ B,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
inhabited-rat-cube_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
istype-less_than,
rat-cube-dimension_wf,
rational-cube_wf,
istype-nat,
istype-void,
istype-le,
sum_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
rat-interval-dimension_wf,
int_seg_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_subtype_base,
lelt_wf,
int_subtype_base,
primrec-wf2,
less_than_wf,
equal-wf-base,
squash_wf,
true_wf,
sum_split1,
decidable__lt,
subtype_rel_self,
iff_weakening_equal,
non_neg_sum,
int_seg_properties,
subtype_rel_function,
rational-interval_wf,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
int_seg_subtype_special,
int_seg_cases
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
sqequalRule,
imageElimination,
voidElimination,
natural_numberEquality,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
universeIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
functionIsType,
productIsType,
equalityIstype,
intEquality,
baseClosed,
sqequalBase,
setIsType,
functionEquality,
productEquality,
imageMemberEquality,
universeEquality,
applyLambdaEquality,
addEquality,
minusEquality,
multiplyEquality,
hypothesis_subsumption
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k). (0 < dim(c) {}\mRightarrow{} (\mexists{}i:\mBbbN{}k. (dim(c i) = 1)))
Date html generated:
2020_05_20-AM-09_19_21
Last ObjectModification:
2019_11_13-PM-06_13_07
Theory : rationals
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