Nuprl Lemma : simple-fan-theorem
∀[X:(𝔹 List) ⟶ ℙ]. ∀bar:tbar(𝔹;X). ∀d: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],
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
member: t ∈ T,
nat: ℕ,
so_apply: x[s1;s2],
uimplies: b supposing a,
tbar: tbar(T;X),
squash: ↓T,
implies: P ⇒ Q,
prop: ℙ,
guard: {T},
dec-predicate: Decidable(X)
Lemmas referenced :
tbar_wf,
list_wf,
dec-predicate_wf,
nat_wf,
upto_wf,
bool_wf,
int_seg_wf,
map_wf,
simple_fan_theorem
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
hypothesis,
functionEquality,
independent_isectElimination,
introduction,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
cumulativity,
universeEquality
Latex:
\mforall{}[X:(\mBbbB{} List) {}\mrightarrow{} \mBbbP{}]
\mforall{}bar:tbar(\mBbbB{};X). \mforall{}d:Decidable(X). (\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_15
Last ObjectModification:
2016_01_16-PM-04_05_30
Theory : bar!induction
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