Nuprl Lemma : complement-unbounded-tree
∀[T:Type]. ∀A:{A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ 𝔹| Tree(A) ∧ Unbounded(A)} . (¬bar(¬(A))) supposing ¬¬Fan(T)
Proof
Definitions occuring in Statement :
altneg: ¬(A),
alttree: Tree(A),
altunbounded: Unbounded(A),
altfan: Fan(T),
altbar: bar(X),
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
nat: ℕ,
prop: ℙ,
squash: ↓T,
cand: A c∧ B,
and: P ∧ Q,
false: False,
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-void,
altbar_wf,
altfan_wf,
altunbounded_wf,
iff_weakening_equal,
subtype_rel_self,
altneg-altneg,
istype-universe,
bool_wf,
int_seg_wf,
istype-nat,
true_wf,
squash_wf,
alttree_wf,
altneg_wf,
fan-bar-not-unbounded
Rules used in proof :
Error :inhabitedIsType,
Error :functionIsTypeImplies,
Error :productIsType,
Error :setIsType,
voidElimination,
independent_pairFormation,
baseClosed,
imageMemberEquality,
sqequalRule,
because_Cache,
universeEquality,
instantiate,
natural_numberEquality,
Error :functionIsType,
Error :universeIsType,
equalitySymmetry,
equalityTransitivity,
imageElimination,
Error :lambdaEquality_alt,
applyEquality,
promote_hyp,
productElimination,
independent_functionElimination,
dependent_functionElimination,
rename,
setElimination,
Error :lambdaFormation_alt,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[T:Type]. \mforall{}A:\{A:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} \mBbbB{}| Tree(A) \mwedge{} Unbounded(A)\} . (\mneg{}bar(\mneg{}(A))) supposing \mneg{}\mneg{}Fan(T)
Date html generated:
2019_06_20-PM-02_46_40
Last ObjectModification:
2019_06_07-AM-11_16_26
Theory : fan-theorem
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