Nuprl Lemma : length-rat-cube-faces
∀k:ℕ. ∀c:ℚCube(k). ((↑Inhabited(c)) ⇒ (||rat-cube-faces(k;c)|| = (2 * dim(c)) ∈ ℤ))
Proof
Definitions occuring in Statement :
rat-cube-faces: rat-cube-faces(k;c),
rat-cube-dimension: dim(c),
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k),
length: ||as||,
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rat-cube-faces: rat-cube-faces(k;c),
mapfilter: mapfilter(f;P;L),
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
rational-cube: ℚCube(k),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
compose: f o g,
lsum: Σ(f[x] | x ∈ L),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
rat-cube-dimension: dim(c),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
length-concat,
map_wf,
int_seg_wf,
list_wf,
top_wf,
cons_wf,
rational-cube_wf,
lower-rc-face_wf,
upper-rc-face_wf,
nil_wf,
subtype_rel_list,
istype-void,
filter_wf5,
upto_wf,
l_member_wf,
eq_int_wf,
rat-interval-dimension_wf,
rat-cube-dimension_wf,
subtype_rel_self,
iff_weakening_equal,
map-map,
length_of_cons_lemma,
length_of_nil_lemma,
istype-assert,
inhabited-rat-cube_wf,
istype-nat,
lsum-constant,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
length-filter-lsum,
lsum-upto,
ifthenelse_wf,
sum_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
inhabitedIsType,
instantiate,
universeEquality,
intEquality,
closedConclusion,
natural_numberEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
productElimination,
setIsType,
multiplyEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
productIsType,
equalityElimination,
equalityIstype,
promote_hyp,
cumulativity,
applyLambdaEquality
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k). ((\muparrow{}Inhabited(c)) {}\mRightarrow{} (||rat-cube-faces(k;c)|| = (2 * dim(c))))
Date html generated:
2020_05_20-AM-09_21_59
Last ObjectModification:
2019_11_27-AM-10_16_35
Theory : rationals
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