Nuprl Lemma : decidable__exists-unit-ball-approx-1
∀k,n:ℕ. ∀[P:unit-ball-approx(n;k) ⟶ ℙ]. ((∀p:unit-ball-approx(n;k). Dec(P[p])) ⇒ Dec(∃p:unit-ball-approx(n;k). P[p]))
Proof
Definitions occuring in Statement :
unit-ball-approx: unit-ball-approx(n;k),
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
so_apply: x[s],
prop: ℙ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
unit-ball-approx: unit-ball-approx(n;k),
subtract: n - m,
sq_type: SQType(T),
guard: {T},
cand: A c∧ B,
sq_stable: SqStable(P),
true: True,
extend-approx-ball: extend-approx-ball(n;p;z),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
unit-ball-approx_wf,
istype-void,
istype-le,
decidable_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-less_than,
primrec-wf2,
istype-nat,
unit-ball-approx0,
subtype_rel_self,
int_seg_wf,
le_wf,
sum_wf,
int_seg_properties,
extend-approx-ball_wf,
subtype_base_sq,
int_subtype_base,
add-associates,
add-swap,
add-commutes,
zero-add,
decidable__exists_int_seg,
decidable__cand,
unit-ball-approx-subtype,
decidable__lt,
sq_stable__le,
sum-unroll,
istype-top,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
isect_memberFormation_alt,
sqequalRule,
functionIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
voidElimination,
hypothesis,
hypothesisEquality,
applyEquality,
universeEquality,
rename,
setElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
isectIsType,
because_Cache,
productEquality,
setIsType,
instantiate,
isectEquality,
functionEquality,
cumulativity,
inhabitedIsType,
productElimination,
inlFormation_alt,
productIsType,
inrFormation_alt,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
minusEquality,
addEquality,
imageElimination,
multiplyEquality,
equalityTransitivity,
intEquality,
imageMemberEquality,
baseClosed,
lessCases,
axiomSqEquality,
isectIsTypeImplies,
functionExtensionality,
equalityElimination,
equalityIstype,
promote_hyp
Latex:
\mforall{}k,n:\mBbbN{}.
\mforall{}[P:unit-ball-approx(n;k) {}\mrightarrow{} \mBbbP{}]
((\mforall{}p:unit-ball-approx(n;k). Dec(P[p])) {}\mRightarrow{} Dec(\mexists{}p:unit-ball-approx(n;k). P[p]))
Date html generated:
2019_10_30-AM-11_28_22
Last ObjectModification:
2019_07_30-AM-11_30_01
Theory : real!vectors
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