Nuprl Lemma : RankEx2-induction
∀[S,T:Type]. ∀[P:RankEx2(S;T) ─→ ℙ].
((∀leaft:T. P[RankEx2_LeafT(leaft)])
⇒ (∀leafs:S. P[RankEx2_LeafS(leafs)])
⇒ (∀prod:RankEx2(S;T) × S × T. (let u,u1 = prod in let u1,u2 = u in P[u1] ⇒ P[RankEx2_Prod(prod)]))
⇒ (∀union:S × RankEx2(S;T) + RankEx2(S;T)
(case union of inl(u) => let u1,u2 = u in P[u2] | inr(u1) => P[u1] ⇒ P[RankEx2_Union(union)]))
⇒ (∀listprod:(S × RankEx2(S;T)) List. ((∀u∈listprod.let u1,u2 = u in P[u2]) ⇒ P[RankEx2_ListProd(listprod)]))
⇒ (∀unionlist:T + (RankEx2(S;T) List)
(case unionlist of inl(u) => True | inr(u1) => (∀u∈u1.P[u]) ⇒ P[RankEx2_UnionList(unionlist)]))
⇒ {∀v:RankEx2(S;T). P[v]})
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},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
true: True,
function: x:A ─→ B[x],
spread: spread def,
product: x:A × B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
union: left + right,
universe: Type
Lemmas :
uniform-comp-nat-induction,
all_wf,
isect_wf,
le_wf,
RankEx2_size_wf,
nat_wf,
less_than_wf,
RankEx2-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
decidable__lt,
false_wf,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
subtract_wf,
decidable__le,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-zero,
subtract-is-less,
lelt_wf,
sum-nat,
length_wf_nat,
select_wf,
sq_stable__le,
int_seg_wf,
length_wf,
sum_wf,
RankEx2_wf,
sum-nat-less,
uall_wf,
le_weakening,
list_wf,
true_wf,
l_all_wf2,
l_member_wf,
RankEx2_UnionList_wf,
RankEx2_ListProd_wf,
RankEx2_Union_wf,
RankEx2_Prod_wf,
RankEx2_LeafS_wf,
RankEx2_LeafT_wf
\mforall{}[S,T:Type]. \mforall{}[P:RankEx2(S;T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}leaft:T. P[RankEx2\_LeafT(leaft)])
{}\mRightarrow{} (\mforall{}leafs:S. P[RankEx2\_LeafS(leafs)])
{}\mRightarrow{} (\mforall{}prod:RankEx2(S;T) \mtimes{} S \mtimes{} T
(let u,u1 = prod in let u1,u2 = u in P[u1] {}\mRightarrow{} P[RankEx2\_Prod(prod)]))
{}\mRightarrow{} (\mforall{}union:S \mtimes{} RankEx2(S;T) + RankEx2(S;T)
(case union of inl(u) => let u1,u2 = u in P[u2] | inr(u1) => P[u1]
{}\mRightarrow{} P[RankEx2\_Union(union)]))
{}\mRightarrow{} (\mforall{}listprod:(S \mtimes{} RankEx2(S;T)) List
((\mforall{}u\mmember{}listprod.let u1,u2 = u in P[u2]) {}\mRightarrow{} P[RankEx2\_ListProd(listprod)]))
{}\mRightarrow{} (\mforall{}unionlist:T + (RankEx2(S;T) List)
(case unionlist of inl(u) => True | inr(u1) => (\mforall{}u\mmember{}u1.P[u])
{}\mRightarrow{} P[RankEx2\_UnionList(unionlist)]))
{}\mRightarrow{} \{\mforall{}v:RankEx2(S;T). P[v]\})
Date html generated:
2015_07_17-AM-07_50_15
Last ObjectModification:
2015_01_27-AM-09_38_08
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