Nuprl Lemma : not-tree-big
∀[T:Type]. ∀[A:(T List) ⟶ ℙ].
((∃k:ℕ. T ~ ℕk)
⇒ Decidable(A)
⇒ (∀n:ℕ. ((¬tree-big(T;upwd-closure(T;A);n)) ⇒ (∃as:T List. ((||as|| = n ∈ ℤ) ∧ (¬(upwd-closure(T;A) as)))))))
Proof
Definitions occuring in Statement :
tree-big: tree-big(T;A;n),
upwd-closure: upwd-closure(T;A),
dec-predicate: Decidable(X),
equipollent: A ~ B,
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
tree-big: tree-big(T;A;n),
cand: A c∧ B
Lemmas referenced :
not_wf,
tree-big_wf,
upwd-closure_wf,
nat_wf,
dec-predicate_wf,
list_wf,
exists_wf,
equipollent_wf,
int_seg_wf,
decidable__exists_length,
decidable__not,
decidable__upwd-closure,
not_over_exists,
and_wf,
equal_wf,
length_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
functionEquality,
cumulativity,
universeEquality,
applyEquality,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
unionElimination,
intEquality,
productElimination,
independent_isectElimination,
voidElimination,
independent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[A:(T List) {}\mrightarrow{} \mBbbP{}].
((\mexists{}k:\mBbbN{}. T \msim{} \mBbbN{}k)
{}\mRightarrow{} Decidable(A)
{}\mRightarrow{} (\mforall{}n:\mBbbN{}
((\mneg{}tree-big(T;upwd-closure(T;A);n))
{}\mRightarrow{} (\mexists{}as:T List. ((||as|| = n) \mwedge{} (\mneg{}(upwd-closure(T;A) as)))))))
Date html generated:
2016_05_14-PM-04_10_19
Last ObjectModification:
2015_12_26-PM-07_54_21
Theory : fan-theorem
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