Nuprl Lemma : cantor-interval-cauchy
∀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 :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
cauchy: cauchy(n.x[n]),
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
nat: ℕ,
subtype_rel: A ⊆r B,
pi1: fst(t),
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
sq_type: SQType(T),
subtract: n - m,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
true: True,
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
int_upper: {i...},
pi2: snd(t)
Lemmas referenced :
le_witness_for_triv,
cantor-interval-inclusion,
cantor-interval-length,
istype-le,
rleq_wf,
rabs_wf,
rsub_wf,
cantor-interval_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
nat_plus_wf,
istype-nat,
bool_wf,
real_wf,
r-archimedean,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_weakening2,
exp_wf2,
mul-swap,
mul-commutes,
zero-mul,
exp_step,
istype-less_than,
mul_bounds_1b,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
mul_nat_plus,
intformeq_wf,
int_formula_prop_eq_lemma,
log-property,
log_wf,
exp_wf4,
nat_plus_subtype_nat,
le_functionality,
multiply_functionality_wrt_le,
le_weakening,
le_wf,
squash_wf,
true_wf,
exp-of-mul,
subtype_rel_self,
iff_weakening_equal,
itermMultiply_wf,
int_term_value_mul_lemma,
set_subtype_base,
less_than_wf,
exp_mul,
assert_of_le_int,
exp_preserves_le,
istype-false,
int-rdiv_wf,
exp_wf3,
nequal_wf,
int-rmul_wf,
exp-positive-stronger,
rmul_wf,
rleq_functionality,
int-rdiv-req,
req_weakening,
rdiv_functionality,
int-rmul-req,
rmul_preserves_rleq,
rinv_wf2,
req_transitivity,
rmul_functionality,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmul_preserves_rleq2,
rleq-int,
rmul-int,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
req_inversion,
rmul-rinv,
ge_wf,
subtract-1-ge-0,
add-zero,
itermAdd_wf,
int_term_value_add_lemma,
exp_add,
exp_wf_nat_plus,
exp1,
trivial-int-eq1,
multiply-is-int-iff,
false_wf,
subtype_rel_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
rleq_transitivity,
rleq_weakening,
rabs-difference-symmetry,
rleq-implies-rleq,
radd-preserves-rleq,
radd_wf,
rabs-of-nonneg,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
productElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
rename,
because_Cache,
setElimination,
dependent_set_memberEquality_alt,
independent_functionElimination,
functionIsType,
universeIsType,
applyEquality,
equalityIstype,
closedConclusion,
natural_numberEquality,
inrFormation_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
instantiate,
cumulativity,
intEquality,
dependent_set_memberFormation_alt,
multiplyEquality,
imageMemberEquality,
baseClosed,
imageElimination,
universeEquality,
sqequalBase,
intWeakElimination,
addEquality,
applyLambdaEquality,
hyp_replacement,
pointwiseFunctionality,
promote_hyp,
baseApply,
productIsType
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_10
Last ObjectModification:
2019_02_11-PM-02_12_02
Theory : reals
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