Nuprl Lemma : cover-seq-property
∀[A,B:ℝ ⟶ ℙ].
∀d:r:ℝ ⟶ (A[r] + B[r]). ∀a,b:ℝ.
(A[a]
⇒ B[b]
⇒ (∀n:ℕ
(A[fst(cover-seq(d;a;b;n))]
∧ B[snd(cover-seq(d;a;b;n))]
∧ ((cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <a, (a + b/r(2))> ∈ (ℝ × ℝ))
∨ (cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <(a + b/r(2)), b> ∈ (ℝ × ℝ))))))
Proof
Definitions occuring in Statement :
cover-seq: cover-seq(d;a;b;n),
rdiv: (x/y),
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
spread: spread def,
pair: <a, b>,
product: x:A × B[x],
union: left + right,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
cover-seq: cover-seq(d;a;b;n),
member: t ∈ T,
top: Top,
primrec: primrec(n;b;c),
pi1: fst(t),
pi2: snd(t),
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
cand: A c∧ B,
so_lambda: λ2x.t[x],
nat: ℕ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
primrec0_lemma,
primrec1_lemma,
real_wf,
rdiv_wf,
radd_wf,
int-to-real_wf,
rless-int,
rless_wf,
equal_wf,
cover-seq_wf,
subtract_wf,
decidable__le,
full-omega-unsat,
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,
pi1_wf_top,
pi2_wf,
or_wf,
subtract-add-cancel,
set_wf,
less_than_wf,
primrec-wf2,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
primrec-unroll,
subtype_base_sq,
bool_wf,
bool_subtype_base,
squash_wf,
true_wf,
eq_int_eq_false,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
bfalse_wf,
iff_weakening_equal,
and_wf,
subtype_rel_product,
top_wf,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
applyEquality,
functionExtensionality,
hypothesisEquality,
isectElimination,
natural_numberEquality,
independent_isectElimination,
inrFormation,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
unionEquality,
unionElimination,
independent_pairEquality,
productEquality,
inlFormation,
equalityTransitivity,
equalitySymmetry,
rename,
setElimination,
dependent_set_memberEquality,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
addEquality,
functionEquality,
universeEquality,
cumulativity,
instantiate,
imageElimination,
addLevel,
hyp_replacement,
applyLambdaEquality,
levelHypothesis,
baseApply,
closedConclusion
Latex:
\mforall{}[A,B:\mBbbR{} {}\mrightarrow{} \mBbbP{}].
\mforall{}d:r:\mBbbR{} {}\mrightarrow{} (A[r] + B[r]). \mforall{}a,b:\mBbbR{}.
(A[a]
{}\mRightarrow{} B[b]
{}\mRightarrow{} (\mforall{}n:\mBbbN{}
(A[fst(cover-seq(d;a;b;n))]
\mwedge{} B[snd(cover-seq(d;a;b;n))]
\mwedge{} ((cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <a, (a + b/r(2))>)
\mvee{} (cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <(a + b/r(2)), b>)))))
Date html generated:
2017_10_03-AM-10_03_20
Last ObjectModification:
2017_07_06-AM-11_11_16
Theory : reals
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