Nuprl Lemma : cantor-interval-rleq
∀[a,b:ℝ]. ∀[n:ℕ]. ∀[f:ℕn ⟶ 𝔹]. ((fst(cantor-interval(a;b;f;n))) ≤ (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),
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
real: ℝ,
cantor-interval: cantor-interval(a;b;f;n),
pi1: fst(t),
pi2: snd(t),
less_than': less_than'(a;b),
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 ,
int_seg: {i..j-},
lelt: i ≤ j < k,
int_nzero: ℤ-o,
true: True,
sq_stable: SqStable(P),
squash: ↓T,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
less_than'_wf,
rsub_wf,
cantor-interval_wf,
nat_plus_properties,
real_wf,
pi2_wf,
equal_wf,
pi1_wf_top,
nat_plus_wf,
int_seg_wf,
bool_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
le_wf,
nat_wf,
rleq_wf,
primrec0_lemma,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
sq_stable__rleq,
decidable__lt,
lelt_wf,
int-rdiv_wf,
int_subtype_base,
true_wf,
nequal_wf,
radd_wf,
int-rmul_wf,
equal-wf-base,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rmul_preserves_rleq,
rmul_wf,
primrec-unroll,
rleq_functionality,
int-rdiv-req,
req_weakening,
uiff_transitivity,
rmul-rdiv-cancel2,
rmul_comm,
radd_comm,
rmul_preserves_rleq2,
rleq-int,
radd-preserves-rleq,
radd_functionality,
int-rmul-req,
req_transitivity,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
squash_wf,
radd_comm_eq,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination,
independent_pairEquality,
applyEquality,
because_Cache,
productEquality,
equalityTransitivity,
equalitySymmetry,
minusEquality,
axiomEquality,
functionEquality,
unionElimination,
dependent_set_memberEquality,
functionExtensionality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
addLevel,
spreadEquality,
imageMemberEquality,
baseClosed,
imageElimination,
inrFormation,
addEquality,
universeEquality
Latex:
\mforall{}[a,b:\mBbbR{}].
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbB{}]. ((fst(cantor-interval(a;b;f;n))) \mleq{} (snd(cantor-interval(a;b;f;n))))
supposing a \mleq{} b
Date html generated:
2017_10_03-AM-09_50_48
Last ObjectModification:
2017_07_28-AM-08_01_10
Theory : reals
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