Nuprl Lemma : bar-separation-implies-twkl!
∀[T:Type]
((∃size:ℕ. T ~ ℕsize)
⇒ BarSep(T;T)
⇒ (∀A:(T List) ⟶ ℙ. (dbar(T;A) ⇒ (¬(down-closed(T;¬(A)) ∧ Unbounded(¬(A))))))
⇒ WKL!(T))
Proof
Definitions occuring in Statement :
twkl!: WKL!(T),
unbounded-list-predicate: Unbounded(A),
down-closed: down-closed(T;X),
bar-separation: BarSep(T;S),
dbar: dbar(T;X),
predicate-not: ¬(A),
equipollent: A ~ B,
list: T List,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
compose: f o g,
tbar: tbar(T;X),
is-path: is-path(A;f),
sq_exists: ∃x:A [B[x]],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
nil: [],
lt_int: i <z j,
from-upto: [n, m),
upto: upto(n),
subtract: n - m,
pi1: fst(t),
finite-type: finite-type(T),
nat_plus: ℕ+,
label: ...$L... t,
bnot: ¬bb,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
eff-unique-path: eff-unique-path(T;A),
jbar: jbar(T;S;X;Y),
bar-separation: BarSep(T;S),
less_than: a < b,
select: L[n],
l_member: (x ∈ l),
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
inject: Inj(A;B;f),
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
ge: i ≥ j ,
int_seg: {i..j-},
top: Top,
cons: [a / b],
true: True,
sq_type: SQType(T),
less_than': less_than'(a;b),
le: A ≤ B,
surject: Surj(A;B;f),
biject: Bij(A;B;f),
equipollent: A ~ B,
squash: ↓T,
sq_stable: SqStable(P),
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
cand: A c∧ B,
unbounded-list-predicate: Unbounded(A),
predicate-not: ¬(A),
R-closed: R-closed(T;x.X[x];a,b.R[a; b]),
down-closed: down-closed(T;X),
dbar: dbar(T;X),
rev_implies: P ⇐ Q,
or: P ∨ Q,
decidable: Dec(P),
iff: P ⇐⇒ Q,
dec-predicate: Decidable(X),
nat: ℕ,
exists: ∃x:A. B[x],
false: False,
subtype_rel: A ⊆r B,
not: ¬A,
and: P ∧ Q,
prop: ℙ,
member: t ∈ T,
all: ∀x:A. B[x],
twkl!: WKL!(T),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
add-zero,
zero-mul,
add-mul-special,
minus-one-mul,
minus-add,
map-map,
upto_decomp,
add_functionality_wrt_eq,
primrec-wf2,
decidable__equal_list,
is-path_wf,
subtract-add-cancel,
zero-add,
add-commutes,
add-swap,
add-associates,
map_cons_lemma,
map_append_sq,
upto_decomp1,
primrec_wf,
assert_of_le_int,
bnot_of_lt_int,
assert_functionality_wrt_uiff,
bnot_wf,
le_int_wf,
equal-wf-base,
uiff_transitivity,
length-map,
primrec-unroll,
subtract-1-ge-0,
map_nil_lemma,
istype-base,
stuck-spread,
primrec0_lemma,
ge_wf,
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
btrue_wf,
surject_wf,
equipollent_inversion,
finite-type-iff-list,
nat_plus_properties,
add_nat_wf,
add_nat_plus,
select-upto,
map-length,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
select_upto,
false_wf,
add-is-int-iff,
select-append,
length_upto,
top_wf,
subtype_rel_list,
select-map,
le_weakening2,
iseg_select,
istype-false,
int_seg_subtype_nat,
nat_wf,
subtype_rel_function,
upto_wf,
map_wf,
iff_weakening_equal,
select_append_back,
true_wf,
squash_wf,
non_neg_length,
select-cons-hd,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
itermAdd_wf,
itermSubtract_wf,
subtract_wf,
less_than_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
length-singleton,
length-append,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
decidable__not,
cons_member,
subtype_rel_sets_simple,
member_singleton,
select_wf,
length_wf,
eff-unique-path_wf,
istype-assert,
null_cons_lemma,
istype-true,
null_nil_lemma,
nil_wf,
cons_wf,
append_wf,
equal_wf,
l_member_wf,
null_wf,
assert_wf,
not_wf,
list_induction,
decidable__equal_int_seg,
istype-less_than,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformand_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
length_of_cons_lemma,
product_subtype_list,
subtype_base_sq,
length_of_nil_lemma,
list-cases,
istype-le,
sq_stable_from_decidable,
sq_stable__all,
tbar_wf,
int_subtype_base,
le_wf,
set_subtype_base,
length_wf_nat,
istype-int,
iseg_wf,
decidable-predicate-not,
istype-universe,
int_seg_wf,
equipollent_wf,
istype-nat,
bar-separation_wf,
istype-void,
subtype_rel_self,
predicate-not_wf,
dbar_wf,
unbounded-list-predicate_wf,
down-closed_wf,
list_wf,
dec-predicate_wf
Rules used in proof :
multiplyEquality,
Error :dependent_set_memberFormation_alt,
Error :functionIsTypeImplies,
axiomEquality,
independent_pairEquality,
intWeakElimination,
Error :functionExtensionality_alt,
hyp_replacement,
baseApply,
pointwiseFunctionality,
equalityElimination,
closedConclusion,
addEquality,
setEquality,
productEquality,
Error :inrFormation_alt,
Error :inlFormation_alt,
int_eqEquality,
approximateComputation,
applyLambdaEquality,
Error :isect_memberEquality_alt,
hypothesis_subsumption,
cumulativity,
Error :dependent_set_memberEquality_alt,
imageElimination,
baseClosed,
imageMemberEquality,
functionEquality,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
intEquality,
Error :equalityIstype,
Error :dependent_pairFormation_alt,
promote_hyp,
productElimination,
Error :inhabitedIsType,
Error :lambdaEquality_alt,
voidElimination,
independent_functionElimination,
unionElimination,
dependent_functionElimination,
independent_pairFormation,
natural_numberEquality,
instantiate,
applyEquality,
functionExtensionality,
universeEquality,
Error :functionIsType,
Error :productIsType,
sqequalRule,
because_Cache,
Error :setIsType,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
Error :universeIsType,
cut,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]
((\mexists{}size:\mBbbN{}. T \msim{} \mBbbN{}size)
{}\mRightarrow{} BarSep(T;T)
{}\mRightarrow{} (\mforall{}A:(T List) {}\mrightarrow{} \mBbbP{}. (dbar(T;A) {}\mRightarrow{} (\mneg{}(down-closed(T;\mneg{}(A)) \mwedge{} Unbounded(\mneg{}(A))))))
{}\mRightarrow{} WKL!(T))
Date html generated:
2019_06_20-PM-02_48_08
Last ObjectModification:
2019_06_05-PM-04_21_08
Theory : fan-theorem
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