Nuprl Lemma : lg-all-remove
∀[T:Type]. ∀[P:T ─→ ℙ]. ∀g:LabeledGraph(T). ∀x:ℕlg-size(g). (∀x∈g.P[x] ⇐⇒ ∀x∈lg-remove(g;x).P[x] ∧ P[lg-label(g;x)])
Proof
Definitions occuring in Statement :
lg-all: ∀x∈G.P[x],
lg-label: lg-label(g;x),
lg-remove: lg-remove(g;n),
lg-size: lg-size(g),
labeled-graph: LabeledGraph(T),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ─→ B[x],
natural_number: $n,
universe: Type
Lemmas :
int_seg_wf,
lg-size_wf,
lg-remove_wf,
int_seg_subtype-nat,
false_wf,
nat_wf,
all_wf,
lg-label2_wf,
labeled-graph_wf,
int_seg_properties,
lelt_wf,
squash_wf,
true_wf,
lg-size-remove,
iff_weakening_equal,
lg-label-remove,
subtract_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
decidable__lt,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
decidable__le,
not-le-2,
zero-add,
add-zero,
le-add-cancel,
minus-minus,
iff_imp_equal_bool,
btrue_wf,
assert_wf,
iff_wf,
sq_stable__le,
le-add-cancel-alt,
bnot_wf,
not_wf,
int_subtype_base,
lg-label_wf,
and_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
decidable__equal_int,
not-equal-2
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
\mforall{}g:LabeledGraph(T). \mforall{}x:\mBbbN{}lg-size(g). (\mforall{}x\mmember{}g.P[x] \mLeftarrow{}{}\mRightarrow{} \mforall{}x\mmember{}lg-remove(g;x).P[x] \mwedge{} P[lg-label(g;x)])
Date html generated:
2015_07_23-AM-11_04_48
Last ObjectModification:
2015_07_16-AM-10_04_44
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