Nuprl Lemma : simple_fan_theorem
∀[X:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ ℙ]
(∀n:ℕ. ∀s:ℕn ⟶ 𝔹. Dec(X[n;s])) ⇒ (∃k:ℕ [(∀f:ℕ ⟶ 𝔹. ∃n:ℕk. X[n;f])]) supposing ∀f:ℕ ⟶ 𝔹. (↓∃n:ℕ. X[n;f])
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s1;s2],
int_seg: {i..j-},
guard: {T},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
true: True,
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
less_than: a < b,
seq-append: seq-append(n;m;s1;s2),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
ge: i ≥ j ,
nat_plus: ℕ+,
seq-adjoin: s++t
Lemmas referenced :
nat_wf,
bool_wf,
basic_bar_induction,
sq_exists_wf,
all_wf,
exists_wf,
int_seg_wf,
add_nat_wf,
sq_stable__le,
equal_wf,
le_wf,
seq-append_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_function,
subtype_rel_self,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
seq-adjoin_wf,
decidable_wf,
squash_wf,
and_wf,
less_than_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not-lt-2,
less-iff-le,
btrue_wf,
set_wf,
set-value-type,
int-value-type,
bfalse_wf,
add-member-int_seg2,
subtract_wf,
le-add-cancel2,
decidable__lt,
minus-minus,
sq_stable__and,
sq_stable__less_than,
member-less_than,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
minus-zero,
omega-shadow,
mul-distributes,
mul-commutes,
mul-associates,
mul-swap,
nat_properties,
int_subtype_base,
iff_imp_equal_bool,
assert_wf,
true_wf,
not-less-implies-equal,
le-add-cancel-alt,
set_subtype_base,
decidable__int_equal,
not-equal-2,
less_than_transitivity1,
less_than_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
imageElimination,
hypothesis,
imageMemberEquality,
baseClosed,
functionEquality,
extract_by_obid,
rename,
lambdaFormation,
isectElimination,
because_Cache,
natural_numberEquality,
setElimination,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
addEquality,
independent_functionElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_pairFormation,
unionElimination,
voidElimination,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
cumulativity,
universeEquality,
dependent_set_memberFormation,
dependent_pairFormation,
addLevel,
hyp_replacement,
levelHypothesis,
equalityElimination,
lessCases,
axiomSqEquality,
promote_hyp,
instantiate,
cutEval,
multiplyEquality
Latex:
\mforall{}[X:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbP{}]
(\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbB{}. Dec(X[n;s])) {}\mRightarrow{} (\mexists{}k:\mBbbN{} [(\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \mexists{}n:\mBbbN{}k. X[n;f])])
supposing \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. (\mdownarrow{}\mexists{}n:\mBbbN{}. X[n;f])
Date html generated:
2019_06_20-AM-11_32_31
Last ObjectModification:
2018_08_20-PM-09_32_25
Theory : bool_1
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