Nuprl Lemma : vdf-wf
∀[A,B:Type]. ∀[C:A ⟶ B ⟶ Type].
∀n:ℕ
((vdf(A;B;a,b.C[a;b];n) ∈ Type)
∧ (∀f:vdf(A;B;a,b.C[a;b];n). ∀L:(a:A × b:B × C[a;b]) List. ((||L|| ≤ (n + 1)) ⇒ (vdf-eq(A;f;L) ∈ ℙ))))
Proof
Definitions occuring in Statement :
vdf: vdf(A;B;a,b.C[a; b];n),
vdf-eq: vdf-eq(A;f;L),
length: ||as||,
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
so_apply: x[s1;s2],
nat: ℕ
Lemmas referenced :
vdf-wf+,
istype-le,
length_wf,
list_wf,
istype-nat,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
dependent_functionElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
independent_functionElimination,
productEquality,
applyEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
universeIsType,
functionIsType,
inhabitedIsType,
instantiate,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[C:A {}\mrightarrow{} B {}\mrightarrow{} Type].
\mforall{}n:\mBbbN{}
((vdf(A;B;a,b.C[a;b];n) \mmember{} Type)
\mwedge{} (\mforall{}f:vdf(A;B;a,b.C[a;b];n). \mforall{}L:(a:A \mtimes{} b:B \mtimes{} C[a;b]) List.
((||L|| \mleq{} (n + 1)) {}\mRightarrow{} (vdf-eq(A;f;L) \mmember{} \mBbbP{}))))
Date html generated:
2020_05_19-PM-09_40_14
Last ObjectModification:
2020_03_05-PM-04_26_19
Theory : co-recursion-2
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