Nuprl Lemma : cantor-interval-inclusion
∀[a,b:ℝ].
∀[f:ℕ ⟶ 𝔹]. ∀[n:ℕ]. ∀[m:{n...}].
(((fst(cantor-interval(a;b;f;n))) ≤ (fst(cantor-interval(a;b;f;m))))
∧ ((fst(cantor-interval(a;b;f;m))) ≤ (snd(cantor-interval(a;b;f;m))))
∧ ((snd(cantor-interval(a;b;f;m))) ≤ (snd(cantor-interval(a;b;f;n)))))
supposing a ≤ b
Proof
Definitions occuring in Statement :
cantor-interval: cantor-interval(a;b;f;n),
rleq: x ≤ y,
real: ℝ,
int_upper: {i...},
nat: ℕ,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than': less_than'(a;b),
cantor-interval: cantor-interval(a;b;f;n),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
subtract: n - m,
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
rev_uimplies: rev_uimplies(P;Q),
int_nzero: ℤ-o,
rge: x ≥ y,
real: ℝ,
int_upper: {i...}
Lemmas referenced :
nat_wf,
bool_wf,
rleq_wf,
real_wf,
less_than'_wf,
rsub_wf,
cantor-interval_wf,
nat_plus_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
subtype_rel_dep_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
pi1_wf_top,
equal_wf,
nat_plus_wf,
pi2_wf,
primrec-unroll,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
subtract_wf,
add-subtract-cancel,
add-associates,
add-swap,
add-commutes,
zero-add,
assert_elim,
int_subtype_base,
not_assert_elim,
btrue_neq_bfalse,
cantor-interval-rleq,
rmul_preserves_rleq,
rdiv_wf,
rless-int,
rleq_weakening_equal,
rmul_wf,
int-to-real_wf,
radd_wf,
int-rmul_wf,
rless_wf,
int-rdiv_wf,
equal-wf-base,
true_wf,
nequal_wf,
rleq_functionality,
req_weakening,
int-rdiv-req,
uiff_transitivity,
rmul-rdiv-cancel2,
rmul_comm,
radd_comm,
radd_functionality,
int-rmul-req,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
req_transitivity,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul_preserves_rleq2,
rleq-int,
int_upper_subtype_nat,
int_upper_wf,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
less_than_transitivity1,
less_than_irreflexivity,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-zero,
rleq_transitivity,
decidable__equal_int,
squash_wf,
and_wf,
subtype_rel_product,
top_wf,
int_upper_properties,
trivial-int-eq1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
functionEquality,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
applyEquality,
because_Cache,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
lambdaFormation,
productEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
minusEquality,
axiomEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
functionExtensionality,
hyp_replacement,
applyLambdaEquality,
inrFormation,
imageMemberEquality,
baseClosed,
addLevel,
intWeakElimination,
imageElimination
Latex:
\mforall{}[a,b:\mBbbR{}].
\mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbB{}]. \mforall{}[n:\mBbbN{}]. \mforall{}[m:\{n...\}].
(((fst(cantor-interval(a;b;f;n))) \mleq{} (fst(cantor-interval(a;b;f;m))))
\mwedge{} ((fst(cantor-interval(a;b;f;m))) \mleq{} (snd(cantor-interval(a;b;f;m))))
\mwedge{} ((snd(cantor-interval(a;b;f;m))) \mleq{} (snd(cantor-interval(a;b;f;n)))))
supposing a \mleq{} b
Date html generated:
2017_10_03-AM-09_51_27
Last ObjectModification:
2017_07_28-AM-08_01_36
Theory : reals
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