Nuprl Lemma : cantor-interval-length
∀[a,b:ℝ]. ∀[n:ℕ]. ∀[f:ℕn ⟶ 𝔹]. (((snd(cantor-interval(a;b;f;n))) - fst(cantor-interval(a;b;f;n))) = (2^n * b - a)/3^n)
Proof
Definitions occuring in Statement :
cantor-interval: cantor-interval(a;b;f;n),
int-rdiv: (a)/k1,
int-rmul: k1 * a,
rsub: x - y,
req: x = y,
real: ℝ,
exp: i^n,
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
cantor-interval: cantor-interval(a;b;f;n),
decidable: Dec(P),
or: P ∨ Q,
pi2: snd(t),
pi1: fst(t),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
less_than: a < b,
rneq: x ≠ y,
guard: {T},
rless: x < y,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
req_witness,
rsub_wf,
cantor-interval_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
int-rdiv_wf,
exp_wf3,
intformeq_wf,
int_formula_prop_eq_lemma,
nequal_wf,
int-rmul_wf,
exp_wf2,
subtract-1-ge-0,
istype-nat,
real_wf,
int_seg_wf,
int_seg_properties,
req_weakening,
subtype_rel_function,
bool_wf,
subtract_wf,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtype_rel_self,
sq_stable__req,
ifthenelse_wf,
lt_int_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
radd_wf,
exp_step,
exp-positive-stronger,
rless-int,
nat_plus_properties,
rmul_preserves_req,
rless_wf,
int-to-real_wf,
req_wf,
rdiv_wf,
rmul_wf,
rminus_wf,
itermMultiply_wf,
itermAdd_wf,
itermMinus_wf,
rinv_wf2,
int_term_value_mul_lemma,
rneq_functionality,
rmul-int,
rneq-int,
set_subtype_base,
less_than_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
req-iff-rsub-is-0,
nat_plus_inc_int_nzero,
primrec-unroll,
exp0_lemma,
req_functionality,
req_transitivity,
int-rdiv-one,
int-rmul-one,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
int-rdiv-req,
rsub_functionality,
rdiv_functionality,
int-rmul-req,
radd_functionality,
rmul_functionality,
rmul-rinv3,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
req_inversion,
rminus_functionality,
rinv-mul-as-rdiv,
rinv-of-rmul,
int-rinv-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
dependent_set_memberEquality_alt,
unionElimination,
productElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
because_Cache,
baseClosed,
sqequalBase,
intEquality,
functionIsType,
applyEquality,
addEquality,
minusEquality,
multiplyEquality,
productEquality,
independent_pairEquality,
productIsType,
imageMemberEquality,
imageElimination,
inrFormation_alt,
closedConclusion,
baseApply,
instantiate,
cumulativity,
equalityElimination,
promote_hyp
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbB{}].
(((snd(cantor-interval(a;b;f;n))) - fst(cantor-interval(a;b;f;n))) = (2\^{}n * b - a)/3\^{}n)
Date html generated:
2019_10_30-AM-07_37_38
Last ObjectModification:
2018_12_16-PM-04_42_33
Theory : reals
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