Nuprl Lemma : alt-bar-sep-wkl!
∀[T:Type]
((∃size:ℕ. T ~ ℕsize)
⇒ BarSep(T;T)
⇒ (∀A:{A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ 𝔹| Tree(A) ∧ Unbounded(A)} . (¬bar(¬(A))))
⇒ WKL!(T))
Proof
Definitions occuring in Statement :
alt-wkl!: WKL!(T),
altneg: ¬(A),
altbarsep: BarSep(T;S),
alttree: Tree(A),
altunbounded: Unbounded(A),
altbar: bar(X),
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
altbar: bar(X),
altpath: IsPath(A;f),
sq_exists: ∃x:A [B[x]],
pi1: fst(t),
nat_plus: ℕ+,
rev_uimplies: rev_uimplies(P;Q),
altneg: ¬(A),
seq+: s.t,
bnot: ¬bb,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
seq-append: seq-append(n;s;s'),
alt-one-path: AtMostOnePath(A),
altjbar: jbar(X;Y),
altbarsep: BarSep(T;S),
rev_implies: P ⇐ Q,
cand: A c∧ B,
select: L[n],
l_member: (x ∈ l),
true: True,
inject: Inj(A;B;f),
bfalse: ff,
cons: [a / b],
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
finite-type: finite-type(T),
iff: P ⇐⇒ Q,
surject: Surj(A;B;f),
biject: Bij(A;B;f),
altunbounded: Unbounded(A),
equipollent: A ~ B,
sq_type: SQType(T),
so_apply: x[s],
guard: {T},
sq_stable: SqStable(P),
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
exists: ∃x:A. B[x],
ge: i ≥ j ,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
alttree: Tree(A),
member: t ∈ T,
all: ∀x:A. B[x],
alt-wkl!: WKL!(T),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
iff_weakening_equal,
true_wf,
squash_wf,
primrec-wf2,
not_assert_elim,
assert_elim,
decidable__equal_function,
altpath_wf,
lelt_wf,
set_subtype_base,
subtract-1-ge-0,
ge_wf,
false_wf,
add-is-int-iff,
nat_plus_properties,
length_wf_nat,
add_nat_wf,
add_nat_plus,
bnot_wf,
assert_of_bnot,
int_seg_subtype_nat,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
less_than_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
cons_member,
subtype_rel_sets_simple,
member_singleton,
select_wf,
length_wf,
cons_wf,
length_of_nil_lemma,
length_of_cons_lemma,
istype-true,
list_wf,
equal_wf,
null_wf,
not_wf,
list_induction,
seq+_wf,
seq-append_wf,
int_term_value_add_lemma,
itermAdd_wf,
l_member_wf,
istype-universe,
equipollent_wf,
altbarsep_wf,
altneg_wf,
altbar_wf,
altunbounded_wf,
alttree_wf,
bool_wf,
alt-one-path_wf,
null_cons_lemma,
product_subtype_list,
btrue_neq_bfalse,
nil_wf,
member-implies-null-eq-bfalse,
btrue_wf,
null_nil_lemma,
list-cases,
surject_wf,
equipollent_inversion,
finite-type-iff-list,
decidable__equal_int_seg,
int_formula_prop_eq_lemma,
intformeq_wf,
istype-less_than,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
assert_witness,
decidable__assert,
sq_stable_from_decidable,
istype-assert,
istype-nat,
subtype_rel_self,
le_weakening2,
sq_stable__le,
istype-false,
int_seg_subtype,
subtype_rel_function,
istype-le,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
assert_wf,
int_seg_wf,
nat_wf,
sq_stable__all
Rules used in proof :
Error :functionExtensionality_alt,
axiomEquality,
intWeakElimination,
functionExtensionality,
baseApply,
pointwiseFunctionality,
hyp_replacement,
equalityElimination,
closedConclusion,
setEquality,
productEquality,
addEquality,
universeEquality,
Error :setIsType,
hypothesis_subsumption,
Error :inrFormation_alt,
Error :equalityIstype,
Error :inlFormation_alt,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
Error :productIsType,
intEquality,
cumulativity,
instantiate,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
Error :functionIsType,
baseClosed,
imageMemberEquality,
Error :universeIsType,
independent_pairFormation,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
imageElimination,
productElimination,
Error :dependent_set_memberEquality_alt,
because_Cache,
applyEquality,
natural_numberEquality,
functionEquality,
Error :lambdaEquality_alt,
sqequalRule,
hypothesis,
isectElimination,
extract_by_obid,
introduction,
rename,
setElimination,
cut,
promote_hyp,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
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:\{A:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} \mBbbB{}| Tree(A) \mwedge{} Unbounded(A)\} . (\mneg{}bar(\mneg{}(A))))
{}\mRightarrow{} WKL!(T))
Date html generated:
2019_06_20-PM-02_46_55
Last ObjectModification:
2019_06_07-AM-11_57_39
Theory : fan-theorem
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