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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
labeled-graph: LabeledGraph(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
lg-edge: lg-edge(g;a;b),
lg-append: lg-append(g1;g2),
lg-in-edges: lg-in-edges(g;x),
top: Top,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
lg-size: lg-size(g),
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
pi2: snd(t),
pi1: fst(t),
iff: P ⇐⇒ Q,
l_member: (x ∈ l),
cand: A c∧ B,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
spreadn: spread3
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:
2016_05_17-AM-10_09_54
Last ObjectModification:
2016_01_18-AM-00_23_19
Theory : process-model
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