Nuprl Lemma : fan-implies-barred-not-unbounded
∀[T:Type]. (Fan_d(T) ⇒ (∀A:(T List) ⟶ ℙ. (dbar(T;A) ⇒ (¬(down-closed(T;¬(A)) ∧ Unbounded(¬(A)))))))
Proof
Definitions occuring in Statement :
unbounded-list-predicate: Unbounded(A),
down-closed: down-closed(T;X),
dfan: Fan_d(T),
dbar: dbar(T;X),
predicate-not: ¬(A),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
predicate-not: ¬(A),
true: True,
cand: A c∧ B,
squash: ↓T,
less_than: a < b,
lelt: i ≤ j < k,
int_seg: {i..j-},
R-closed: R-closed(T;x.X[x];a,b.R[a; b]),
down-closed: down-closed(T;X),
less_than': less_than'(a;b),
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
unbounded-list-predicate: Unbounded(A),
exists: ∃x:A. B[x],
ubar: ubar(T;X),
prop: ℙ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
dfan: Fan_d(T),
false: False,
not: ¬A,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
select-upto,
top_wf,
subtype_rel_list,
select-map,
length_wf,
map-length,
int_seg_subtype_nat,
length_upto,
iff_weakening_equal,
istype-nat,
map_length_nat,
true_wf,
squash_wf,
le_wf,
iseg_select,
upto_wf,
int_seg_properties,
int_seg_wf,
map_wf,
istype-false,
istype-less_than,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
intformeq_wf,
intformless_wf,
decidable__lt,
select_wf,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
istype-le,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_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,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
istype-universe,
dfan_wf,
dbar_wf,
unbounded-list-predicate_wf,
subtype_rel_self,
list_wf,
predicate-not_wf,
down-closed_wf
Rules used in proof :
baseClosed,
imageMemberEquality,
imageElimination,
closedConclusion,
cumulativity,
promote_hyp,
Error :equalityIstype,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
independent_pairFormation,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
independent_isectElimination,
unionElimination,
natural_numberEquality,
rename,
setElimination,
addEquality,
Error :dependent_set_memberEquality_alt,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
Error :lambdaEquality_alt,
Error :functionIsType,
universeEquality,
instantiate,
applyEquality,
functionExtensionality,
isectElimination,
extract_by_obid,
Error :universeIsType,
Error :productIsType,
sqequalRule,
voidElimination,
because_Cache,
productElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
dependent_functionElimination,
sqequalHypSubstitution,
thin,
Error :lambdaFormation_alt,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]
(Fan\_d(T) {}\mRightarrow{} (\mforall{}A:(T List) {}\mrightarrow{} \mBbbP{}. (dbar(T;A) {}\mRightarrow{} (\mneg{}(down-closed(T;\mneg{}(A)) \mwedge{} Unbounded(\mneg{}(A)))))))
Date html generated:
2019_06_20-PM-02_47_25
Last ObjectModification:
2019_06_05-PM-04_39_49
Theory : fan-theorem
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