Nuprl Lemma : lg-edge-append
∀[T:Type]
∀g1,g2:LabeledGraph(T). ∀a,b:ℕlg-size(g1) + lg-size(g2).
(lg-edge(lg-append(g1;g2);a;b)
⇐⇒ (a < lg-size(g1) ∧ b < lg-size(g1) ∧ lg-edge(g1;a;b))
∨ ((lg-size(g1) ≤ a) ∧ (lg-size(g1) ≤ b) ∧ lg-edge(g2;a - lg-size(g1);b - lg-size(g1))))
Proof
Definitions occuring in Statement :
lg-edge: lg-edge(g;a;b),
lg-append: lg-append(g1;g2),
lg-size: lg-size(g),
labeled-graph: LabeledGraph(T),
int_seg: {i..j-},
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type
Lemmas :
length_wf_nat,
top_wf,
dep-isect-subtype,
list_wf,
int_seg_wf,
length_wf,
select_wf,
nat_wf,
sq_stable__le,
lg-size_wf,
non_neg_length,
less_than_wf,
l_member_wf,
subtype_rel_list,
less_than_transitivity1,
less_than_irreflexivity,
or_wf,
le_wf,
subtract_wf,
pi1_wf_top,
zero-le-nat,
map_wf,
iff_wf,
exists_wf,
member_map,
lelt_wf,
subtype_base_sq,
int_subtype_base
Latex:
\mforall{}[T:Type]
\mforall{}g1,g2:LabeledGraph(T). \mforall{}a,b:\mBbbN{}lg-size(g1) + lg-size(g2).
(lg-edge(lg-append(g1;g2);a;b)
\mLeftarrow{}{}\mRightarrow{} (a < lg-size(g1) \mwedge{} b < lg-size(g1) \mwedge{} lg-edge(g1;a;b))
\mvee{} ((lg-size(g1) \mleq{} a) \mwedge{} (lg-size(g1) \mleq{} b) \mwedge{} lg-edge(g2;a - lg-size(g1);b - lg-size(g1))))
Date html generated:
2015_07_22-PM-00_28_59
Last ObjectModification:
2015_01_28-PM-11_35_01
Home
Index