Nuprl Lemma : cantor-lemma2
∀z:ℕ ⟶ ℝ
∃f:ℕ ⟶ {p:ℝ × ℝ| (fst(p)) < (snd(p))} ⟶ {p:ℝ × ℝ| (fst(p)) < (snd(p))}
∀n:ℕ. ∀p:{p:ℝ × ℝ| (fst(p)) < (snd(p))} .
let x,y = p
in let x',y' = f n p
in (x ≤ x') ∧ (x' < y') ∧ (y' ≤ y) ∧ (((z n) < x') ∨ (y' < (z n))) ∧ ((y' - x') < (r1/r(n + 1)))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
nat: ℕ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
subtype_rel: A ⊆r B,
spreadn: spread3,
squash: ↓T,
true: True,
sq_stable: SqStable(P)
Lemmas referenced :
cantor-lemma,
rless_wf,
real_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
uimplies_subtype,
rless-int-fractions2,
rleq_wf,
rsub_wf,
squash_wf,
true_wf,
pi1_wf_top,
istype-nat,
or_wf,
sq_stable__rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
rename,
sqequalHypSubstitution,
sqequalRule,
dependent_pairFormation_alt,
lambdaEquality_alt,
isectElimination,
thin,
productElimination,
hypothesisEquality,
hypothesis,
productIsType,
universeIsType,
because_Cache,
setElimination,
applyEquality,
closedConclusion,
natural_numberEquality,
addEquality,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
productEquality,
inhabitedIsType,
unionIsType,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
independent_pairEquality,
equalityIsType1,
setIsType,
functionIsType
Latex:
\mforall{}z:\mBbbN{} {}\mrightarrow{} \mBbbR{}
\mexists{}f:\mBbbN{} {}\mrightarrow{} \{p:\mBbbR{} \mtimes{} \mBbbR{}| (fst(p)) < (snd(p))\} {}\mrightarrow{} \{p:\mBbbR{} \mtimes{} \mBbbR{}| (fst(p)) < (snd(p))\}
\mforall{}n:\mBbbN{}. \mforall{}p:\{p:\mBbbR{} \mtimes{} \mBbbR{}| (fst(p)) < (snd(p))\} .
let x,y = p
in let x',y' = f n p
in (x \mleq{} x')
\mwedge{} (x' < y')
\mwedge{} (y' \mleq{} y)
\mwedge{} (((z n) < x') \mvee{} (y' < (z n)))
\mwedge{} ((y' - x') < (r1/r(n + 1)))
Date html generated:
2019_10_29-AM-10_25_05
Last ObjectModification:
2018_11_08-PM-06_00_57
Theory : reals
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