Nuprl Lemma : rat-cube-dimension-0
∀[k:ℕ]. ∀[c:ℚCube(k)]. uiff(dim(c) = 0 ∈ ℤ;(↑Inhabited(c)) ∧ (∀i:ℕk. (dim(c i) = 0 ∈ ℤ)))
Proof
Definitions occuring in Statement :
rat-cube-dimension: dim(c),
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k),
rat-interval-dimension: dim(I),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
it: ⋅,
unit: Unit,
bool: 𝔹,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
squash: ↓T,
rational-cube: ℚCube(k),
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
nat: ℕ,
false: False,
true: True,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
guard: {T},
implies: P ⇒ Q,
sq_type: SQType(T),
or: P ∨ Q,
all: ∀x:A. B[x],
rat-cube-dimension: dim(c),
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
istype-less_than,
istype-le,
int_formula_prop_less_lemma,
intformless_wf,
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,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
istype-false,
int_seg_subtype_nat,
sum-nat-le-simple,
uiff_transitivity,
not_wf,
bnot_wf,
assert_wf,
equal-wf-T-base,
iff_weakening_equal,
subtype_rel_self,
sum-is-zero,
istype-universe,
true_wf,
squash_wf,
equal_wf,
istype-nat,
rational-cube_wf,
rat-interval-dimension_wf,
istype-assert,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
istype-int,
inhabited-rat-cube_wf,
assert_witness,
int_seg_wf,
int_subtype_base,
assert_of_bnot,
eqff_to_assert,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases
Rules used in proof :
applyLambdaEquality,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
dependent_set_memberEquality_alt,
equalityElimination,
imageMemberEquality,
universeEquality,
imageElimination,
isectIsTypeImplies,
isect_memberEquality_alt,
functionIsType,
productIsType,
sqequalBase,
baseClosed,
addEquality,
minusEquality,
applyEquality,
equalityIstype,
inhabitedIsType,
functionIsTypeImplies,
axiomEquality,
lambdaEquality_alt,
independent_pairEquality,
hypothesisEquality,
rename,
setElimination,
universeIsType,
lambdaFormation_alt,
voidElimination,
natural_numberEquality,
intEquality,
sqequalRule,
productElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
hypothesis,
cumulativity,
isectElimination,
instantiate,
unionElimination,
thin,
dependent_functionElimination,
extract_by_obid,
because_Cache,
sqequalHypSubstitution,
independent_pairFormation,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)]. uiff(dim(c) = 0;(\muparrow{}Inhabited(c)) \mwedge{} (\mforall{}i:\mBbbN{}k. (dim(c i) = 0)))
Date html generated:
2019_10_29-AM-07_52_09
Last ObjectModification:
2019_10_27-PM-01_05_35
Theory : rationals
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