Nuprl Lemma : rat-cube-dimension-zero
∀[k:ℕ]. ∀[c:ℚCube(k)]. uiff(dim(c) = 0 ∈ ℤ;∀i:ℕk. ((fst((c i))) = (snd((c i))) ∈ ℚ))
Proof
Definitions occuring in Statement :
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k),
rationals: ℚ,
int_seg: {i..j-},
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
nat: ℕ,
rev_implies: P ⇐ Q,
not: ¬A,
assert: ↑b,
bnot: ¬bb,
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
false: False,
true: True,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
guard: {T},
prop: ℙ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
pi2: snd(t),
pi1: fst(t),
inhabited-rat-interval: Inhabited(I),
rat-interval-dimension: dim(I),
rational-interval: ℚInterval,
implies: P ⇒ Q,
rational-cube: ℚCube(k),
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rat-cube-dimension-0,
rat-cube-dimension_wf,
equal-wf-base,
assert-inhabited-rat-cube,
inhabited-rat-cube_wf,
assert_wf,
iff_weakening_uiff,
istype-nat,
rational-cube_wf,
rationals_wf,
assert_witness,
qless_irreflexivity,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
lelt_wf,
set_subtype_base,
rat-interval-dimension_wf,
inhabited-rat-interval_wf,
int_seg_wf,
q_le_wf,
istype-assert,
assert-q_le-eq,
qle_antisymmetry,
qless_complement_qorder,
qless_wf,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
qle_wf,
istype-int,
int_subtype_base,
subtype_base_sq,
iff_weakening_equal,
assert-q_less-eq,
eqtt_to_assert,
q_less_wf
Rules used in proof :
productEquality,
addEquality,
minusEquality,
functionEquality,
isectIsTypeImplies,
isect_memberEquality_alt,
independent_pairEquality,
functionIsType,
productIsType,
functionIsTypeImplies,
axiomEquality,
lambdaEquality_alt,
rename,
setElimination,
promote_hyp,
dependent_pairFormation_alt,
universeIsType,
sqequalBase,
baseClosed,
equalityIstype,
voidElimination,
natural_numberEquality,
intEquality,
cumulativity,
instantiate,
independent_functionElimination,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
isectElimination,
extract_by_obid,
sqequalRule,
inhabitedIsType,
applyEquality,
because_Cache,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
independent_pairFormation,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)]. uiff(dim(c) = 0;\mforall{}i:\mBbbN{}k. ((fst((c i))) = (snd((c i)))))
Date html generated:
2019_10_29-AM-07_52_24
Last ObjectModification:
2019_10_27-PM-01_19_58
Theory : rationals
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