Nuprl Lemma : fan-theorem
∀[X:(𝔹 List) ⟶ ℙ]. (tbar(𝔹;X) ⇒ Decidable(X) ⇒ (∃k:ℕ. ∀f:ℕ ⟶ 𝔹. ∃n:ℕk. (X map(f;upto(n)))))
Proof
Definitions occuring in Statement :
tbar: tbar(T;X),
dec-predicate: Decidable(X),
upto: upto(n),
map: map(f;as),
list: T List,
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uiff: uiff(P;Q),
dec-predicate: Decidable(X),
decidable: Dec(P),
or: P ∨ Q,
outl: outl(x)
Lemmas referenced :
dec-predicate_wf,
list_wf,
bool_wf,
tbar_wf,
simple-fan-theorem,
nat_wf,
all_wf,
exists_wf,
int_seg_wf,
map_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
upto_wf,
not_over_exists,
not_wf,
int_seg_decide_wf,
decidable_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
setElimination,
dependent_pairFormation,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
because_Cache,
independent_isectElimination,
independent_pairFormation,
addLevel,
impliesFunctionality,
productElimination,
independent_functionElimination,
voidElimination,
introduction,
instantiate,
equalityTransitivity,
equalitySymmetry,
unionElimination
Latex:
\mforall{}[X:(\mBbbB{} List) {}\mrightarrow{} \mBbbP{}]. (tbar(\mBbbB{};X) {}\mRightarrow{} Decidable(X) {}\mRightarrow{} (\mexists{}k:\mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \mexists{}n:\mBbbN{}k. (X map(f;upto(n)))))
Date html generated:
2016_05_15-PM-10_05_18
Last ObjectModification:
2015_12_27-PM-05_51_09
Theory : bar!induction
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