Nuprl Lemma : closures-meet-sq
∀[P,Q:ℝ ⟶ ℙ].
((∃a:{a:ℝ| P a} . (∃b:ℝ [((Q b) ∧ (a ≤ b))]))
⇒ (∃c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)}
∀a:{a:ℝ| P a} . ∀b:{b:ℝ| (Q b) ∧ (a ≤ b)} .
∃a':{a':ℝ| P a'} . (∃b':{b':ℝ| (Q b') ∧ (a' ≤ b')} [((a ≤ a') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))]))
⇒ (∃y:ℝ. (y ∈ closure(λz.(↓P z)) ∧ y ∈ closure(λz.(↓Q z)))))
Proof
Definitions occuring in Statement :
member-closure: y ∈ closure(A),
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
rbetween: x≤y≤z,
rless: x < y,
nat_plus: ℕ+,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
real: ℝ,
nequal: a ≠ b ∈ T ,
rmul: a * b,
member-closure: y ∈ closure(A)
Lemmas referenced :
real_wf,
rleq_wf,
int-to-real_wf,
rless_wf,
subtype_rel_self,
rsub_wf,
rmul_wf,
sq_stable__rleq,
pi1_wf_top,
istype-void,
pi2_wf,
primrec_wf,
int_seg_wf,
istype-nat,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
intformless_wf,
int_formula_prop_less_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
add-subtract-cancel,
sq_stable__all,
nat_wf,
sq_stable__and,
le_witness_for_triv,
ge_wf,
rnexp_zero_lemma,
subtract-1-ge-0,
radd_wf,
rminus_wf,
itermSubtract_wf,
itermMinus_wf,
nat_plus_properties,
itermMultiply_wf,
rleq_weakening_equal,
rleq_functionality,
req_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
radd-preserves-rleq,
subtract_wf,
int_term_value_subtract_lemma,
subtract-add-cancel,
rleq-implies-rleq,
rnexp_wf,
rleq_functionality_wrt_implies,
rmul_preserves_rleq2,
ifthenelse_wf,
eq_int_wf,
sq_stable__less_than,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
neg_assert_of_eq_int,
rmul_functionality,
rnexp-req,
rbetween_wf,
rmul-limit,
constant-limit,
rpowers-converge-ext,
rabs_wf,
sq_stable__rless,
rless_functionality,
rabs-of-nonneg,
converges-to_functionality,
common-limit-squeeze-ext,
member-closure_wf,
squash_wf,
converges-to_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
setElimination,
rename,
cut,
sqequalRule,
productIsType,
setIsType,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
natural_numberEquality,
hypothesisEquality,
functionIsType,
because_Cache,
applyEquality,
instantiate,
universeEquality,
inhabitedIsType,
dependent_pairEquality_alt,
dependent_set_memberEquality_alt,
dependent_functionElimination,
dependent_pairFormation_alt,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
independent_pairEquality,
isect_memberEquality_alt,
voidElimination,
setEquality,
lambdaEquality_alt,
productEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_set_memberFormation_alt,
equalityIstype,
addEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
int_eqEquality,
functionExtensionality,
equalityElimination,
cumulativity,
functionIsTypeImplies,
intWeakElimination
Latex:
\mforall{}[P,Q:\mBbbR{} {}\mrightarrow{} \mBbbP{}].
((\mexists{}a:\{a:\mBbbR{}| P a\} . (\mexists{}b:\mBbbR{} [((Q b) \mwedge{} (a \mleq{} b))]))
{}\mRightarrow{} (\mexists{}c:\{c:\mBbbR{}| (r0 \mleq{} c) \mwedge{} (c < r1)\}
\mforall{}a:\{a:\mBbbR{}| P a\} . \mforall{}b:\{b:\mBbbR{}| (Q b) \mwedge{} (a \mleq{} b)\} .
\mexists{}a':\{a':\mBbbR{}| P a'\} . (\mexists{}b':\{b':\mBbbR{}| (Q b') \mwedge{} (a' \mleq{} b')\} [((a \mleq{} a') \mwedge{} (b' \mleq{} b) \mwedge{} ((b' - a') \mleq{} ((\000Cb - a) * c)))]))
{}\mRightarrow{} (\mexists{}y:\mBbbR{}. (y \mmember{} closure(\mlambda{}z.(\mdownarrow{}P z)) \mwedge{} y \mmember{} closure(\mlambda{}z.(\mdownarrow{}Q z)))))
Date html generated:
2019_10_29-AM-10_41_53
Last ObjectModification:
2019_01_08-PM-00_27_56
Theory : reals
Home
Index