Nuprl Lemma : fan-bar-not-unbounded
∀[T:Type]. ∀A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ 𝔹. (bar(A) ⇒ (¬(Tree(¬(A)) ∧ Unbounded(¬(A))))) supposing ¬¬Fan(T)
Proof
Definitions occuring in Statement :
altneg: ¬(A),
alttree: Tree(A),
altunbounded: Unbounded(A),
altfan: Fan(T),
altbar: bar(X),
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀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 :
true: True,
sq_stable: SqStable(P),
subtract: n - m,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
altneg: ¬(A),
le: A ≤ B,
squash: ↓T,
less_than: a < b,
alttree: Tree(A),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
lelt: i ≤ j < k,
int_seg: {i..j-},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
prop: ℙ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
altunbounded: Unbounded(A),
exists: ∃x:A. B[x],
altubar: uniformBar(X),
and: P ∧ Q,
altfan: Fan(T),
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
true_wf,
squash_wf,
subtype_rel_self,
le-add-cancel,
add_functionality_wrt_le,
less-iff-le,
sq_stable__le,
add-associates,
add-commutes,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
istype-false,
int_seg_subtype,
subtype_rel_function,
assert_of_bnot,
int_seg_properties,
istype-universe,
altfan_wf,
int_seg_wf,
istype-nat,
altbar_wf,
altunbounded_wf,
altneg_wf,
alttree_wf,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
istype-less_than,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
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
Rules used in proof :
functionExtensionality,
hyp_replacement,
baseClosed,
imageMemberEquality,
minusEquality,
imageElimination,
universeEquality,
Error :isectIsTypeImplies,
Error :functionIsTypeImplies,
Error :functionIsType,
cumulativity,
instantiate,
Error :equalityIstype,
equalitySymmetry,
equalityTransitivity,
Error :productIsType,
applyEquality,
equalityElimination,
Error :inhabitedIsType,
because_Cache,
Error :universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
approximateComputation,
independent_isectElimination,
unionElimination,
isectElimination,
extract_by_obid,
natural_numberEquality,
rename,
setElimination,
addEquality,
Error :dependent_set_memberEquality_alt,
productElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
promote_hyp,
independent_functionElimination,
sqequalHypSubstitution,
thin,
Error :lambdaFormation_alt,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]
\mforall{}A:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} \mBbbB{}. (bar(A) {}\mRightarrow{} (\mneg{}(Tree(\mneg{}(A)) \mwedge{} Unbounded(\mneg{}(A))))) supposing \mneg{}\mneg{}Fan(T)
Date html generated:
2019_06_20-PM-02_46_38
Last ObjectModification:
2019_06_07-AM-11_14_59
Theory : fan-theorem
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