Nuprl Lemma : twkl!-implies-dfan
∀T:Type. ((∃k:ℕ. T ~ ℕk) ⇒ WKL!(T) ⇒ Fan_d(T))
Proof
Definitions occuring in Statement :
twkl!: WKL!(T),
dfan: Fan_d(T),
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
dfan: Fan_d(T),
dbar: dbar(T;X),
and: P ∧ Q,
tbar: tbar(T;X),
dec-predicate: Decidable(X),
ubar: ubar(T;X),
member: t ∈ T,
uall: ∀[x:A]. B[x],
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
exists: ∃x:A. B[x],
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
top: Top,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
pi1: fst(t),
compose: f o g,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
equipollent: A ~ B,
biject: Bij(A;B;f),
surject: Surj(A;B;f),
twkl!: WKL!(T),
down-closed: down-closed(T;X),
R-closed: R-closed(T;x.X[x];a,b.R[a; b]),
upwd-closure: upwd-closure(T;A),
let: let,
unbounded-list-predicate: Unbounded(A),
label: ...$L... t,
uiff: uiff(P;Q),
tree-big: tree-big(T;A;n),
eff-unique-path: eff-unique-path(T;A),
sq_exists: ∃x:{A| B[x]},
is-path: is-path(A;f),
sq_stable: SqStable(P),
compat: l1 || l2
Lemmas referenced :
nil_wf,
tree-big-monotone,
dbar_wf,
list_wf,
twkl!_wf,
exists_wf,
nat_wf,
equipollent_wf,
int_seg_wf,
false_wf,
le_wf,
lelt_wf,
map_nil_lemma,
map_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
upto_wf,
all_wf,
tree-big-least,
not-tree-big,
set_wf,
tree-big_wf,
upwd-closure_wf,
not_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
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,
equal_wf,
decidable__tree-big,
decidable__upwd-closure,
length_wf,
compose_wf,
less_than_wf,
squash_wf,
true_wf,
iff_weakening_equal,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
subtype_base_sq,
int_subtype_base,
add_nat_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__equal_int,
non_neg_length,
equipollent-zero,
decidable__equal_equipollent,
length_wf_nat,
let_wf,
append_wf,
subtract_wf,
iseg_wf,
iseg_transitivity,
iseg_append_iff,
length-append,
map-length,
length_upto,
le_weakening2,
length_append,
subtype_rel_list,
top_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
list_extensionality,
length-map,
select-map,
iseg_select,
and_wf,
decidable__exists_length,
decidable__not,
not_over_exists,
down-closed_wf,
unbounded-list-predicate_wf,
decidable__implies,
decidable__and2,
decidable__equal_list,
imax_ub,
imax_wf,
iseg-map,
int_seg_subtype,
upto_iseg,
select_wf,
select-upto,
sq_stable_from_decidable,
decidable__exists_int_seg,
iseg_weakening,
iseg_length,
common_iseg_compat,
iseg_same_length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
cut,
hypothesis,
dependent_functionElimination,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
unionElimination,
functionExtensionality,
applyEquality,
functionEquality,
universeEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
setElimination,
dependent_pairFormation,
dependent_set_memberEquality,
independent_pairFormation,
because_Cache,
imageMemberEquality,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_functionElimination,
promote_hyp,
setEquality,
productEquality,
addEquality,
int_eqEquality,
intEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
instantiate,
addLevel,
levelHypothesis,
hyp_replacement,
inlFormation,
inrFormation
Latex:
\mforall{}T:Type. ((\mexists{}k:\mBbbN{}. T \msim{} \mBbbN{}k) {}\mRightarrow{} WKL!(T) {}\mRightarrow{} Fan\_d(T))
Date html generated:
2017_04_17-AM-09_38_44
Last ObjectModification:
2017_02_27-PM-05_43_56
Theory : fan-theorem
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