Nuprl Lemma : cantor-interval-rless
∀a,b:ℝ. ((a < b) ⇒ (∀n:ℕ. ∀f:ℕn ⟶ 𝔹. ((fst(cantor-interval(a;b;f;n))) < (snd(cantor-interval(a;b;f;n))))))
Proof
Definitions occuring in Statement :
cantor-interval: cantor-interval(a;b;f;n),
rless: x < y,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_apply: x[s],
cantor-interval: cantor-interval(a;b;f;n),
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
real: ℝ,
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,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b
Lemmas referenced :
all_wf,
int_seg_wf,
subtract_wf,
bool_wf,
rless_wf,
cantor-interval_wf,
sq_stable__less_than,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
real_wf,
pi1_wf_top,
equal_wf,
pi2_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_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,
decidable__lt,
lelt_wf,
int-rdiv_wf,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
radd_wf,
int-rmul_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rmul_preserves_rless,
rmul_wf,
primrec-unroll,
rless_functionality,
int-rdiv-req,
req_weakening,
rmul_comm,
rmul-rdiv-cancel2,
radd_comm,
radd-preserves-rless,
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,
lambdaFormation,
cut,
thin,
rename,
setElimination,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
functionEquality,
natural_numberEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
dependent_set_memberEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productEquality,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
applyEquality,
addEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
addLevel,
inrFormation,
levelHypothesis,
universeEquality
Latex:
\mforall{}a,b:\mBbbR{}.
((a < b)
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbB{}. ((fst(cantor-interval(a;b;f;n))) < (snd(cantor-interval(a;b;f;n))))))
Date html generated:
2017_10_03-AM-09_51_06
Last ObjectModification:
2017_07_28-AM-08_01_22
Theory : reals
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