Nuprl Lemma : lg-edge-remove
∀[T:Type]
∀g:LabeledGraph(T). ∀i:ℕlg-size(g). ∀a,b:ℕlg-size(g) - 1.
(lg-edge(lg-remove(g;i);a;b) ⇐⇒ lg-edge(g;if a <z i then a else a + 1 fi ;if b <z i then b else b + 1 fi ))
Proof
Definitions occuring in Statement :
lg-edge: lg-edge(g;a;b),
lg-remove: lg-remove(g;n),
lg-size: lg-size(g),
labeled-graph: LabeledGraph(T),
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type
Lemmas :
select-map,
append_wf,
top_wf,
firstn_wf,
dep-isect-subtype,
list_wf,
int_seg_wf,
length_wf,
nth_tl_wf,
subtract_wf,
lg-size_wf,
nat_wf,
labeled-graph_wf,
length-map-sq,
less_than_wf,
squash_wf,
true_wf,
lg-size-remove,
iff_weakening_equal,
lelt_wf,
lg-remove_wf,
int_seg_subtype-nat,
false_wf,
length_wf_nat,
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,
length_firstn,
subtype_rel_sets,
le_wf,
le_weakening2,
select_wf,
sq_stable__le,
less_than_transitivity2,
select_append_front,
select_firstn,
non_neg_length,
length_firstn_eq,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
zero-le-nat,
add-mul-special,
zero-mul,
minus-minus,
select_append_back,
select_nth_tl,
add_functionality_wrt_eq,
length_nth_tl,
decidable__lt,
less-iff-le,
filter_wf5,
subtype_rel_list,
l_member_wf,
bnot_wf,
eq_int_wf,
le_int_wf,
assert_of_le_int,
map_wf,
iff_wf,
exists_wf,
member_map,
assert_wf,
member_filter,
int_subtype_base,
not_wf,
equal-wf-base-T,
not-equal-2,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
le-add-cancel2,
or_wf,
pi1_wf_top,
pi2_wf,
subtype_rel_product,
subtype_top
Latex:
\mforall{}[T:Type]
\mforall{}g:LabeledGraph(T). \mforall{}i:\mBbbN{}lg-size(g). \mforall{}a,b:\mBbbN{}lg-size(g) - 1.
(lg-edge(lg-remove(g;i);a;b)
\mLeftarrow{}{}\mRightarrow{} lg-edge(g;if a <z i then a else a + 1 fi ;if b <z i then b else b + 1 fi ))
Date html generated:
2015_07_22-PM-00_28_51
Last ObjectModification:
2015_02_04-PM-04_05_24
Home
Index