Nuprl Lemma : RankEx2-defop
∀[T,S,P:Type]. ∀[R:P ─→ RankEx2(S;T) ─→ ℙ].
((∀t:T. (∃x:{P| (R x RankEx2_LeafT(t))}))
⇒ (∀s:S. (∃x:{P| (R x RankEx2_LeafS(s))}))
⇒ (∀d:RankEx2(S;T). ∀s:S. ∀t:T. ((∃x:{P| (R x d)}) ⇒ (∃x:{P| (R x RankEx2_Prod(<<d, s>, t>))})))
⇒ (∀z:S × RankEx2(S;T) + RankEx2(S;T)
(case z of inl(p) => ∃x:{P| (R x (snd(p)))} | inr(d) => ∃x:{P| (R x d)} ⇒ (∃x:{P| (R x RankEx2_Union(z))})))
⇒ (∀L:(S × RankEx2(S;T)) List. ((∀p∈L.∃x:{P| (R x (snd(p)))}) ⇒ (∃x:{P| (R x RankEx2_ListProd(L))})))
⇒ (∀z:T + (RankEx2(S;T) List)
(case z of inl(p) => True | inr(L) => (∀p∈L.∃x:{P| (R x p)}) ⇒ (∃x:{P| (R x RankEx2_UnionList(z))})))
⇒ {∀t:RankEx2(S;T). (∃x:{P| (R x t)})})
Proof
Definitions occuring in Statement :
RankEx2_UnionList: RankEx2_UnionList(unionlist),
RankEx2_ListProd: RankEx2_ListProd(listprod),
RankEx2_Union: RankEx2_Union(union),
RankEx2_Prod: RankEx2_Prod(prod),
RankEx2_LeafS: RankEx2_LeafS(leafs),
RankEx2_LeafT: RankEx2_LeafT(leaft),
RankEx2: RankEx2(S;T),
l_all: (∀x∈L.P[x]),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
pi2: snd(t),
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
union: left + right,
universe: Type
Lemmas :
RankEx2-definition,
sq_exists_wf,
RankEx2_wf,
l_all_wf2,
l_member_wf,
list_wf,
select_wf,
sq_stable__le,
int_seg_wf,
length_wf
\mforall{}[T,S,P:Type]. \mforall{}[R:P {}\mrightarrow{} RankEx2(S;T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}t:T. (\mexists{}x:\{P| (R x RankEx2\_LeafT(t))\}))
{}\mRightarrow{} (\mforall{}s:S. (\mexists{}x:\{P| (R x RankEx2\_LeafS(s))\}))
{}\mRightarrow{} (\mforall{}d:RankEx2(S;T). \mforall{}s:S. \mforall{}t:T. ((\mexists{}x:\{P| (R x d)\}) {}\mRightarrow{} (\mexists{}x:\{P| (R x RankEx2\_Prod(<<d, s>, t>))\}))\000C)
{}\mRightarrow{} (\mforall{}z:S \mtimes{} RankEx2(S;T) + RankEx2(S;T)
(case z of inl(p) => \mexists{}x:\{P| (R x (snd(p)))\} | inr(d) => \mexists{}x:\{P| (R x d)\}
{}\mRightarrow{} (\mexists{}x:\{P| (R x RankEx2\_Union(z))\})))
{}\mRightarrow{} (\mforall{}L:(S \mtimes{} RankEx2(S;T)) List
((\mforall{}p\mmember{}L.\mexists{}x:\{P| (R x (snd(p)))\}) {}\mRightarrow{} (\mexists{}x:\{P| (R x RankEx2\_ListProd(L))\})))
{}\mRightarrow{} (\mforall{}z:T + (RankEx2(S;T) List)
(case z of inl(p) => True | inr(L) => (\mforall{}p\mmember{}L.\mexists{}x:\{P| (R x p)\})
{}\mRightarrow{} (\mexists{}x:\{P| (R x RankEx2\_UnionList(z))\})))
{}\mRightarrow{} \{\mforall{}t:RankEx2(S;T). (\mexists{}x:\{P| (R x t)\})\})
Date html generated:
2015_07_17-AM-07_50_33
Last ObjectModification:
2015_01_27-AM-09_37_01
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