Nuprl Lemma : simple_general_fan_theorem
∀[T:Type]
(Bounded(T)
⇒ (∀[X:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ]
(∀n:ℕ. ∀s:ℕn ⟶ T. Dec(X[n;s])) ⇒ (∃k:ℕ [(∀f:ℕ ⟶ T. ∃n:ℕk. X[n;f])]) supposing ∀f:ℕ ⟶ T. (↓∃n:ℕ. X[n;f])))
Proof
Definitions occuring in Statement :
bounded-type: Bounded(T),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
nat_plus: ℕ+,
ge: i ≥ j ,
seq-adjoin: s++t,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
cand: A c∧ B,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
seq-append: seq-append(n;m;s1;s2),
istype: istype(T),
sq_type: SQType(T),
less_than: a < b,
true: True,
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
prop: ℙ,
so_apply: x[s],
not: ¬A,
false: False,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
sq_stable: SqStable(P),
guard: {T},
int_seg: {i..j-},
so_apply: x[s1;s2],
nat: ℕ,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
bounded-type: Bounded(T)
Lemmas referenced :
less_than_irreflexivity,
less_than_transitivity1,
not-equal-2,
decidable__int_equal,
nat_properties,
le-add-cancel-alt,
mul-commutes,
mul-associates,
mul-distributes,
omega-shadow,
mul-distributes-right,
two-mul,
zero-mul,
add-mul-special,
one-mul,
minus-zero,
le_reflexive,
and_wf,
add-is-int-iff,
istype-sqequal,
le_weakening2,
minus-minus,
decidable__lt,
le-add-cancel2,
subtract_wf,
add-member-int_seg2,
decidable__exists_int_seg,
sq_stable_from_decidable,
sq_stable__all,
less-iff-le,
not-lt-2,
istype-assert,
assert_of_bnot,
iff_weakening_uiff,
less_than_wf,
not_wf,
bnot_wf,
assert_wf,
iff_transitivity,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
istype-top,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
subtype_base_sq,
subtype_rel_dep_function,
istype-less_than,
istype-universe,
squash_wf,
decidable_wf,
istype-nat,
seq-adjoin_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
decidable__le,
istype-void,
subtype_rel_self,
subtype_rel_function,
istype-false,
int_seg_subtype_nat,
seq-append_wf,
istype-le,
sq_stable__le,
add_nat_wf,
int_seg_wf,
exists_wf,
all_wf,
nat_wf,
sq_exists_wf,
basic_bar_induction
Rules used in proof :
baseApply,
multiplyEquality,
functionExtensionality,
promote_hyp,
isectIsTypeImplies,
axiomSqEquality,
lessCases,
equalityElimination,
functionExtensionality_alt,
intEquality,
cumulativity,
hyp_replacement,
dependent_pairFormation_alt,
dependent_set_memberFormation_alt,
universeEquality,
productEquality,
minusEquality,
isect_memberEquality_alt,
unionElimination,
voidElimination,
productIsType,
setIsType,
instantiate,
functionIsType,
universeIsType,
independent_pairFormation,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
productElimination,
independent_functionElimination,
addEquality,
dependent_set_memberEquality_alt,
applyEquality,
setElimination,
natural_numberEquality,
closedConclusion,
because_Cache,
functionEquality,
isectElimination,
extract_by_obid,
rename,
inhabitedIsType,
functionIsTypeImplies,
baseClosed,
imageMemberEquality,
hypothesis,
imageElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality_alt,
sqequalHypSubstitution,
introduction,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[T:Type]
(Bounded(T)
{}\mRightarrow{} (\mforall{}[X:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} \mBbbP{}]
(\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T. Dec(X[n;s])) {}\mRightarrow{} (\mexists{}k:\mBbbN{} [(\mforall{}f:\mBbbN{} {}\mrightarrow{} T. \mexists{}n:\mBbbN{}k. X[n;f])])
supposing \mforall{}f:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}n:\mBbbN{}. X[n;f])))
Date html generated:
2019_10_15-AM-10_20_24
Last ObjectModification:
2019_10_07-PM-04_44_24
Theory : bar-induction
Home
Index