Nuprl Lemma : cantor-interval-cauchy-ext
∀a,b:ℝ. ∀[f:ℕ ⟶ 𝔹]. cauchy(n.fst(cantor-interval(a;b;f;n))) supposing a ≤ b
Proof
Definitions occuring in Statement :
cantor-interval: cantor-interval(a;b;f;n),
cauchy: cauchy(n.x[n]),
rleq: x ≤ y,
real: ℝ,
nat: ℕ,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
member: t ∈ T,
rsub: x - y,
radd: a + b,
accelerate: accelerate(k;f),
reg-seq-list-add: reg-seq-list-add(L),
cons: [a / b],
rminus: -(x),
nil: [],
it: ⋅,
canonical-bound: canonical-bound(r),
absval: |i|,
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cantor_cauchy: cantor_cauchy(a;b;k),
cantor-interval-cauchy,
r-archimedean,
decidable__equal_int,
canonical-bound-property,
decidable__int_equal,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
true: True,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
cantor-interval-cauchy,
lifting-strict-int_eq,
istype-void,
strict4-decide,
lifting-strict-callbyvalue,
value-type-has-value,
int-value-type,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
strict4-divide,
cbv_sqequal,
lifting-strict-less,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
r-archimedean,
decidable__equal_int,
canonical-bound-property,
decidable__int_equal
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueAdd,
baseApply,
closedConclusion,
hypothesisEquality,
productElimination,
intEquality,
universeIsType,
addExceptionCases,
exceptionSqequal,
inrFormation_alt,
imageMemberEquality,
imageElimination,
inlFormation_alt,
because_Cache,
callbyvalueIntEq,
int_eqExceptionCases,
callbyvalueReduce,
sqequalSqle,
divergentSqle,
callbyvalueLess,
inhabitedIsType,
unionElimination,
equalityElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isectIsTypeImplies,
natural_numberEquality,
independent_functionElimination,
sqleReflexivity,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
cumulativity,
lessExceptionCases,
axiomSqleEquality,
exceptionLess
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbB{}]. cauchy(n.fst(cantor-interval(a;b;f;n))) supposing a \mleq{} b
Date html generated:
2019_10_30-AM-07_38_56
Last ObjectModification:
2019_04_02-AM-10_56_00
Theory : reals
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