Nuprl Lemma : RankEx1-induction
∀[T:Type]. ∀[P:RankEx1(T) ─→ ℙ].
((∀leaf:T. P[RankEx1_Leaf(leaf)])
⇒ (∀prod:RankEx1(T) × RankEx1(T). (let u,u1 = prod in P[u] ∧ P[u1] ⇒ P[RankEx1_Prod(prod)]))
⇒ (∀prodl:T × RankEx1(T). (let u,u1 = prodl in P[u1] ⇒ P[RankEx1_ProdL(prodl)]))
⇒ (∀prodr:RankEx1(T) × T. (let u,u1 = prodr in P[u] ⇒ P[RankEx1_ProdR(prodr)]))
⇒ (∀list:RankEx1(T) List. ((∀u∈list.P[u]) ⇒ P[RankEx1_List(list)]))
⇒ {∀v:RankEx1(T). P[v]})
Proof
Definitions occuring in Statement :
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,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ─→ B[x],
spread: spread def,
product: x:A × B[x],
universe: Type
Lemmas :
uniform-comp-nat-induction,
all_wf,
isect_wf,
le_wf,
RankEx1_size_wf,
nat_wf,
less_than_wf,
RankEx1-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,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
subtract-is-less,
lelt_wf,
decidable__lt,
sum-nat,
length_wf_nat,
select_wf,
sq_stable__le,
int_seg_wf,
length_wf,
sum_wf,
RankEx1_wf,
sum-nat-less,
uall_wf,
le_weakening,
list_wf,
l_all_wf2,
l_member_wf,
RankEx1_List_wf,
RankEx1_ProdR_wf,
RankEx1_ProdL_wf,
RankEx1_Prod_wf,
RankEx1_Leaf_wf
\mforall{}[T:Type]. \mforall{}[P:RankEx1(T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}leaf:T. P[RankEx1\_Leaf(leaf)])
{}\mRightarrow{} (\mforall{}prod:RankEx1(T) \mtimes{} RankEx1(T). (let u,u1 = prod in P[u] \mwedge{} P[u1] {}\mRightarrow{} P[RankEx1\_Prod(prod)]))
{}\mRightarrow{} (\mforall{}prodl:T \mtimes{} RankEx1(T). (let u,u1 = prodl in P[u1] {}\mRightarrow{} P[RankEx1\_ProdL(prodl)]))
{}\mRightarrow{} (\mforall{}prodr:RankEx1(T) \mtimes{} T. (let u,u1 = prodr in P[u] {}\mRightarrow{} P[RankEx1\_ProdR(prodr)]))
{}\mRightarrow{} (\mforall{}list:RankEx1(T) List. ((\mforall{}u\mmember{}list.P[u]) {}\mRightarrow{} P[RankEx1\_List(list)]))
{}\mRightarrow{} \{\mforall{}v:RankEx1(T). P[v]\})
Date html generated:
2015_07_17-AM-07_48_32
Last ObjectModification:
2015_01_27-AM-09_38_57
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