Nuprl Lemma : rat-cube-dimension-one
∀k:ℕ. ∀c:ℚCube(k).
(dim(c) = 1 ∈ ℤ ⇐⇒ ∃i:ℕk. (fst((c i)) < snd((c i)) ∧ (∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((fst((c j))) = (snd((c j))) ∈ ℚ)))))
Proof
Definitions occuring in Statement :
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k),
qless: r < s,
rationals: ℚ,
int_seg: {i..j-},
nat: ℕ,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
decidable: Dec(P),
inhabited-rat-interval: Inhabited(I),
true: True,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
guard: {T},
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rat-interval-dimension: dim(I),
cand: A c∧ B,
uiff: uiff(P;Q),
prop: ℙ,
pi2: snd(t),
pi1: fst(t),
rational-interval: ℚInterval,
rev_implies: P ⇐ Q,
false: False,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
not: ¬A,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
rational-cube: ℚCube(k),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
qless_irreflexivity,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qle_weakening_lt_qorder,
decidable__equal_int,
q_le_wf,
assert-q_le-eq,
qle_antisymmetry,
qless_complement_qorder,
qle_wf,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
iff_weakening_equal,
assert-q_less-eq,
eqtt_to_assert,
q_less_wf,
rev_implies_wf,
rat-cube-dimension-1,
not_wf,
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,
qless_wf,
istype-void,
int_subtype_base,
lelt_wf,
set_subtype_base,
rat-interval-dimension_wf,
istype-int,
inhabited-rat-interval_wf,
istype-assert,
int_seg_wf
Rules used in proof :
voidElimination,
cumulativity,
instantiate,
equalityElimination,
unionElimination,
dependent_pairFormation_alt,
productEquality,
addEquality,
minusEquality,
promote_hyp,
functionEquality,
independent_functionElimination,
dependent_functionElimination,
equalityTransitivity,
inhabitedIsType,
imageElimination,
productElimination,
equalitySymmetry,
sqequalBase,
baseClosed,
independent_isectElimination,
lambdaEquality_alt,
intEquality,
equalityIstype,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
universeIsType,
functionIsType,
productIsType,
sqequalRule,
independent_pairFormation,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k).
(dim(c) = 1
\mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}k. (fst((c i)) < snd((c i)) \mwedge{} (\mforall{}j:\mBbbN{}k. ((\mneg{}(j = i)) {}\mRightarrow{} ((fst((c j))) = (snd((c j))))))))
Date html generated:
2019_10_29-AM-07_52_31
Last ObjectModification:
2019_10_27-PM-10_49_27
Theory : rationals
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