Nuprl Lemma : MultiTree-induction
∀[T:Type]. ∀[P:MultiTree(T) ─→ ℙ].
((∀labels:{L:Atom List| 0 < ||L||} . ∀children:{a:Atom| (a ∈ labels)} ─→ MultiTree(T).
((∀u:{a:Atom| (a ∈ labels)} . P[children u]) ⇒ P[MTree_Node(labels;children)]))
⇒ (∀val:T. P[MTree_Leaf(val)])
⇒ {∀v:MultiTree(T). P[v]})
Proof
Definitions occuring in Statement :
MTree_Leaf: MTree_Leaf(val),
MTree_Node: MTree_Node(labels;children),
MultiTree: MultiTree(T),
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 :
uniform-comp-nat-induction,
all_wf,
MultiTree_wf,
isect_wf,
le_wf,
MultiTree_size_wf,
nat_wf,
less_than_wf,
MultiTree-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
sum-nat,
length_wf_nat,
select_wf,
l_member_wf,
list-subtype,
sq_stable__le,
int_seg_wf,
length_wf,
decidable__lt,
sum_wf,
false_wf,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
sum-nat-less,
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,
nat_properties,
set_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
uall_wf,
le_weakening,
MTree_Leaf_wf,
list_wf,
MTree_Node_wf,
and_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{}u:\{a:Atom| (a \mmember{} labels)\} . P[children u]) {}\mRightarrow{} P[MTree\_Node(labels;children)]))
{}\mRightarrow{} (\mforall{}val:T. P[MTree\_Leaf(val)])
{}\mRightarrow{} \{\mforall{}v:MultiTree(T). P[v]\})
Date html generated:
2015_07_17-AM-07_46_13
Last ObjectModification:
2015_01_27-AM-09_46_05
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