Nuprl Lemma : lg-connected-remove
∀[T:Type]
∀g:LabeledGraph(T). ∀i:ℕlg-size(g). ∀a,b:ℕlg-size(g) - 1.
(lg-connected(lg-remove(g;i);a;b)
⇒ lg-connected(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-connected: lg-connected(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],
implies: P ⇒ Q,
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type
Lemmas :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
decidable__lt,
false_wf,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
decidable__le,
not-le-2,
sq_stable__le,
zero-add,
add-zero,
le-add-cancel,
lg-edge_wf,
infix_ap_wf,
int_seg_wf,
rel_exp_wf,
le_wf,
lg-remove_wf,
int_seg_subtype-nat,
subtype_rel_dep_function,
subtype_rel-int_seg,
le_weakening,
subtract_wf,
lg-size_wf,
nat_wf,
subtype_rel_self,
lg-edge-remove,
and_wf,
member_wf,
rel_exp_iff,
less_than_transitivity1,
le_antisymmetry_iff,
minus-minus,
equal-wf-base,
int_subtype_base,
exists_wf,
less_than_irreflexivity,
le_weakening2,
squash_wf,
true_wf,
all_wf,
nat_plus_wf
Latex:
\mforall{}[T:Type]
\mforall{}g:LabeledGraph(T). \mforall{}i:\mBbbN{}lg-size(g). \mforall{}a,b:\mBbbN{}lg-size(g) - 1.
(lg-connected(lg-remove(g;i);a;b)
{}\mRightarrow{} lg-connected(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_29_09
Last ObjectModification:
2015_01_28-PM-11_36_09
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