Nuprl Lemma : cantor-interval-inclusion2
∀[a,b:ℝ].
∀[m:ℕ]. ∀[n:ℕm]. ∀[f:ℕm ⟶ 𝔹].
(((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_seg: {i..j-},
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],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
prop: ℙ,
bfalse: ff,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_upper: {i...},
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
squash: ↓T,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
cantor-interval-inclusion,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
lelt_wf,
equal_wf,
int_seg_subtype_nat,
false_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
rleq_wf,
squash_wf,
true_wf,
pi1_wf_top,
cantor-interval_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
int_seg_wf,
pi2_wf,
nat_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
lambdaEquality,
setElimination,
rename,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
sqequalRule,
productElimination,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
hyp_replacement,
imageElimination,
instantiate,
cumulativity,
imageMemberEquality,
baseClosed,
functionEquality
Latex:
\mforall{}[a,b:\mBbbR{}].
\mforall{}[m:\mBbbN{}]. \mforall{}[n:\mBbbN{}m]. \mforall{}[f:\mBbbN{}m {}\mrightarrow{} \mBbbB{}].
(((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_42
Last ObjectModification:
2017_07_28-AM-08_01_49
Theory : reals
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