Nuprl Lemma : cantor-to-interval-onto-lemma
∀a,b:ℝ.
∀x:ℝ. ∀n:ℕ. ∀f:{f:ℕn ⟶ 𝔹| x ∈ [fst(cantor-interval(a;b;f;n)), snd(cantor-interval(a;b;f;n))]} .
∃g:{g:ℕn + 1 ⟶ 𝔹| x ∈ [fst(cantor-interval(a;b;g;n + 1)), snd(cantor-interval(a;b;g;n + 1))]} . (g = f ∈ (ℕn ⟶ 𝔹))\000C
supposing a < b
Proof
Definitions occuring in Statement :
cantor-interval: cantor-interval(a;b;f;n),
rccint: [l, u],
i-member: r ∈ I,
rless: x < y,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uimplies: b supposing a,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
pi1: fst(t),
pi2: snd(t),
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
ge: i ≥ j ,
cantor-interval: cantor-interval(a;b;f;n),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
i-member: r ∈ I,
rccint: [l, u]
Lemmas referenced :
cantor-middle-third-lemma,
rless_wf,
real_wf,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
istype-less_than,
subtype_rel_self,
nat_plus_properties,
guard_wf,
all_wf,
bool_wf,
isect_wf,
i-member_wf,
rccint_wf,
cantor-interval_wf,
sq_exists_wf,
equal_wf,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
sq_stable__less_than,
itermAdd_wf,
int_term_value_add_lemma,
primrec-wf2,
nat_properties,
istype-nat,
sq_stable__i-member,
pi1_wf_top,
pi2_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
member_rccint_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
primrec-unroll,
add-subtract-cancel,
subtype_rel_product,
top_wf,
cantor-interval-rless,
btrue_wf,
not-equal-2,
rleq_wf,
ifthenelse_wf,
lt_int_wf,
int-rdiv_wf,
nequal_wf,
radd_wf,
int-rmul_wf,
assert_of_lt_int,
iff_weakening_uiff,
assert_wf,
less_than_wf,
bfalse_wf,
sq_stable__rleq,
subtype_rel_set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
universeIsType,
isectElimination,
hypothesis,
inhabitedIsType,
natural_numberEquality,
setElimination,
rename,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
unionElimination,
applyEquality,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
functionIsType,
functionEquality,
closedConclusion,
equalityIstype,
addEquality,
productEquality,
minusEquality,
multiplyEquality,
imageMemberEquality,
baseClosed,
imageElimination,
setIsType,
equalityElimination,
promote_hyp,
dependent_set_memberFormation_alt,
independent_pairEquality,
spreadEquality,
sqequalBase,
dependent_pairEquality_alt,
functionExtensionality,
hyp_replacement
Latex:
\mforall{}a,b:\mBbbR{}.
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}. \mforall{}f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbB{}| x \mmember{} [fst(cantor-interval(a;b;f;n)), snd(cantor-interval(a;b;f;n))]\} .
\mexists{}g:\{g:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbB{}| x \mmember{} [fst(cantor-interval(a;b;g;n + 1)), snd(cantor-interval(a;b;g;n + 1))]\}
(g = f)
supposing a < b
Date html generated:
2019_10_30-AM-07_40_35
Last ObjectModification:
2018_12_11-AM-11_12_08
Theory : reals
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