Nuprl Lemma : rat-cube-dimension-1
∀k:ℕ. ∀c:ℚCube(k).
uiff(dim(c) = 1 ∈ ℤ;(↑Inhabited(c)) ∧ (∃i:ℕk. ((dim(c i) = 1 ∈ ℤ) ∧ (∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ (dim(c j) = 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),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
ge: i ≥ j ,
rat-interval-dimension: dim(I),
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
less_than': less_than'(a;b),
decidable: Dec(P),
prop: ℙ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
sum_aux: sum_aux(k;v;i;x.f[x]),
sum: Σ(f[x] | x < k),
it: ⋅,
unit: Unit,
bool: 𝔹,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
not: ¬A,
rational-cube: ℚCube(k),
exists: ∃x:A. B[x],
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
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),
uall: ∀[x:A]. B[x],
or: P ∨ Q,
rat-cube-dimension: dim(c),
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
all: ∀x:A. B[x]
Lemmas referenced :
neg_assert_of_eq_int,
assert-bnot,
bool_cases_sqequal,
assert_of_eq_int,
eq_int_wf,
ifthenelse_wf,
sum-is-zero,
nat_properties,
istype-universe,
true_wf,
squash_wf,
int_seg_subtype_nat,
Error :isolate_summand2,
int_seg_cases,
int_seg_subtype_special,
iff_weakening_equal,
equal_wf,
false_wf,
add-is-int-iff,
subtype_rel_self,
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,
istype-false,
int_seg_subtype,
subtype_rel_function,
int_term_value_add_lemma,
itermAdd_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
sum-nat-le-simple,
decidable__lt,
decidable__equal_int,
istype-top,
sum-unroll,
equal-wf-base,
primrec-wf2,
istype-less_than,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
istype-le,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
sum_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
int_seg_properties,
uiff_transitivity,
not_wf,
bnot_wf,
assert_wf,
equal-wf-T-base,
istype-nat,
rational-cube_wf,
istype-void,
rat-interval-dimension_wf,
int_seg_wf,
inhabited-rat-cube_wf,
istype-assert,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
istype-int,
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 :
universeEquality,
hypothesis_subsumption,
promote_hyp,
pointwiseFunctionality,
multiplyEquality,
applyLambdaEquality,
closedConclusion,
baseApply,
imageMemberEquality,
isectIsTypeImplies,
axiomSqEquality,
lessCases,
productEquality,
functionEquality,
setIsType,
dependent_set_memberEquality_alt,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
equalityElimination,
inhabitedIsType,
imageElimination,
functionIsType,
universeIsType,
productIsType,
sqequalBase,
baseClosed,
setElimination,
addEquality,
minusEquality,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
equalityIstype,
voidElimination,
natural_numberEquality,
intEquality,
sqequalRule,
productElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
cumulativity,
isectElimination,
instantiate,
unionElimination,
dependent_functionElimination,
extract_by_obid,
because_Cache,
sqequalHypSubstitution,
rename,
thin,
hypothesis,
axiomEquality,
introduction,
cut,
isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k).
uiff(dim(c) = 1;(\muparrow{}Inhabited(c))
\mwedge{} (\mexists{}i:\mBbbN{}k. ((dim(c i) = 1) \mwedge{} (\mforall{}j:\mBbbN{}k. ((\mneg{}(j = i)) {}\mRightarrow{} (dim(c j) = 0))))))
Date html generated:
2019_10_29-AM-07_52_17
Last ObjectModification:
2019_10_27-PM-10_36_38
Theory : rationals
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