Nuprl Lemma : simple_fan_theorem-ext
∀[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 :
member: t ∈ T,
bottom: ⊥,
seq-normalize: seq-normalize(n;s),
uall: ∀[x:A]. B[x],
top: Top,
has-value: (a)↓,
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
simple_fan_theorem,
basic_bar_induction,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
strict4: strict4(F)
Lemmas referenced :
simple_fan_theorem,
strictness-apply,
bottom_diverge,
exception-not-bottom,
has-value_wf_base,
is-exception_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
bottom-sqle,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
exception-not-value,
value-type-has-value,
int-value-type,
lifting-strict-less,
base_wf,
basic_bar_induction
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalSqle,
sqleRule,
sqleReflexivity,
divergentSqle,
callbyvalueCallbyvalue,
callbyvalueReduce,
independent_functionElimination,
callbyvalueExceptionCases,
axiomSqleEquality,
exceptionSqequal,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
callbyvalueLess,
productElimination,
lambdaFormation,
unionElimination,
equalityElimination,
because_Cache,
independent_isectElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
imageElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
cumulativity,
lessExceptionCases,
intEquality,
callbyvalueAdd,
addExceptionCases,
inlFormation
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:
2018_05_21-PM-00_03_36
Last ObjectModification:
2018_05_19-AM-07_10_58
Theory : bool_1
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