Nuprl Lemma : MTree-induction2
∀[T:Type]. ∀[P:MultiTree(T) ─→ ℙ].
((∀labels:{L:Atom List| 0 < ||L||} . ∀children:{a:Atom| (a ∈ labels)} ─→ MultiTree(T).
((∀a∈labels.P[children a]) ⇒ P[MTree_Node(labels;children)]))
⇒ (∀val:T. P[MTree_Leaf(val)])
⇒ {∀x:MultiTree(T). P[x]})
Proof
Definitions occuring in Statement :
MTree_Leaf: MTree_Leaf(val),
MTree_Node: MTree_Node(labels;children),
MultiTree: MultiTree(T),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
length: ||as||,
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ─→ B[x],
natural_number: $n,
atom: Atom,
universe: Type
Lemmas :
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
decidable__equal_int,
subtype_rel-int_seg,
false_wf,
le_weakening,
subtract_wf,
int_seg_properties,
le_wf,
all_wf,
MultiTree_wf,
MTree-rank_wf,
nat_wf,
decidable__lt,
not-equal-2,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
add-commutes,
add-associates,
add_functionality_wrt_le,
zero-add,
le-add-cancel-alt,
less-iff-le,
le-add-cancel,
lelt_wf,
set_wf,
less_than_wf,
primrec-wf2,
decidable__le,
not-le-2,
sq_stable__le,
add-zero,
add-mul-special,
zero-mul,
MultiTree-induction,
MTree_Node_wf,
l_member_wf,
list_wf,
length_wf,
l_all_iff,
zero-le-nat,
list-subtype,
map_wf,
imax-list-ub,
subtype_rel_list,
map-length,
l_exists_iff,
member_map,
l_member-settype,
MTree_Leaf_wf
\mforall{}[T:Type]. \mforall{}[P:MultiTree(T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}labels:\{L:Atom List| 0 < ||L||\} . \mforall{}children:\{a:Atom| (a \mmember{} labels)\} {}\mrightarrow{} MultiTree(T).
((\mforall{}a\mmember{}labels.P[children a]) {}\mRightarrow{} P[MTree\_Node(labels;children)]))
{}\mRightarrow{} (\mforall{}val:T. P[MTree\_Leaf(val)])
{}\mRightarrow{} \{\mforall{}x:MultiTree(T). P[x]\})
Date html generated:
2015_07_17-AM-07_46_35
Last ObjectModification:
2015_01_27-AM-09_47_11
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