Nuprl Lemma : RankEx1_ind_wf
∀[T,A:Type]. ∀[R:A ─→ RankEx1(T) ─→ ℙ]. ∀[v:RankEx1(T)]. ∀[Leaf:leaf:T ─→ {x:A| R[x;RankEx1_Leaf(leaf)]} ].
∀[Prod:prod:(RankEx1(T) × RankEx1(T))
─→ let u,u1 = prod
in {x:A| R[x;u]} ∧ {x:A| R[x;u1]}
─→ {x:A| R[x;RankEx1_Prod(prod)]} ]. ∀[ProdL:prodl:(T × RankEx1(T))
─→ let u,u1 = prodl
in {x:A| R[x;u1]}
─→ {x:A| R[x;RankEx1_ProdL(prodl)]} ].
∀[ProdR:prodr:(RankEx1(T) × T) ─→ let u,u1 = prodr in {x:A| R[x;u]} ─→ {x:A| R[x;RankEx1_ProdR(prodr)]} ].
∀[List:list:(RankEx1(T) List) ─→ (∀u∈list.{x:A| R[x;u]} ) ─→ {x:A| R[x;RankEx1_List(list)]} ].
(RankEx1_ind(v;
RankEx1_Leaf(leaf)⇒ Leaf[leaf];
RankEx1_Prod(prod)⇒ rec1.Prod[prod;rec1];
RankEx1_ProdL(prodl)⇒ rec2.ProdL[prodl;rec2];
RankEx1_ProdR(prodr)⇒ rec3.ProdR[prodr;rec3];
RankEx1_List(list)⇒ rec4.List[list;rec4]) ∈ {x:A| R[x;v]} )
Proof
Definitions occuring in Statement :
RankEx1_ind: RankEx1_ind,
RankEx1_List: RankEx1_List(list),
RankEx1_ProdR: RankEx1_ProdR(prodr),
RankEx1_ProdL: RankEx1_ProdL(prodl),
RankEx1_Prod: RankEx1_Prod(prod),
RankEx1_Leaf: RankEx1_Leaf(leaf),
RankEx1: RankEx1(T),
l_all: (∀x∈L.P[x]),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ─→ B[x],
spread: spread def,
product: x:A × B[x],
universe: Type
Lemmas :
top_wf,
has-value_wf_base,
lifting-strict-atom_eq,
base_wf,
RankEx1_wf,
RankEx1_Leaf_wf,
RankEx1_Prod_wf,
RankEx1_ProdL_wf,
RankEx1_ProdR_wf,
list_wf,
l_all_wf2,
l_member_wf,
set_wf,
RankEx1_List_wf,
all_wf,
subtype_rel-equal
\mforall{}[T,A:Type]. \mforall{}[R:A {}\mrightarrow{} RankEx1(T) {}\mrightarrow{} \mBbbP{}]. \mforall{}[v:RankEx1(T)]. \mforall{}[Leaf:leaf:T {}\mrightarrow{} \{x:A|
R
[x;RankEx1\_Leaf(leaf)]\} ]\000C.
\mforall{}[Prod:prod:(RankEx1(T) \mtimes{} RankEx1(T))
{}\mrightarrow{} let u,u1 = prod
in \{x:A| R[x;u]\} \mwedge{} \{x:A| R[x;u1]\}
{}\mrightarrow{} \{x:A| R[x;RankEx1\_Prod(prod)]\} ]. \mforall{}[ProdL:prodl:(T \mtimes{} RankEx1(T))
{}\mrightarrow{} let u,u1 = prodl
in \{x:A| R[x;u1]\}
{}\mrightarrow{} \{x:A| R[x;RankEx1\_ProdL(prodl)]\} ].
\mforall{}[ProdR:prodr:(RankEx1(T) \mtimes{} T)
{}\mrightarrow{} let u,u1 = prodr
in \{x:A| R[x;u]\}
{}\mrightarrow{} \{x:A| R[x;RankEx1\_ProdR(prodr)]\} ]. \mforall{}[List:list:(RankEx1(T) List)
{}\mrightarrow{} (\mforall{}u\mmember{}list.\{x:A| R[x;u]\} )
{}\mrightarrow{} \{x:A| R[x;RankEx1\_List(list)]\} ].
(RankEx1\_ind(v;
RankEx1\_Leaf(leaf){}\mRightarrow{} Leaf[leaf];
RankEx1\_Prod(prod){}\mRightarrow{} rec1.Prod[prod;rec1];
RankEx1\_ProdL(prodl){}\mRightarrow{} rec2.ProdL[prodl;rec2];
RankEx1\_ProdR(prodr){}\mRightarrow{} rec3.ProdR[prodr;rec3];
RankEx1\_List(list){}\mRightarrow{} rec4.List[list;rec4]) \mmember{} \{x:A| R[x;v]\} )
Date html generated:
2015_07_17-AM-07_48_38
Last ObjectModification:
2015_01_29-PM-04_39_36
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