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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
lg-edge: lg-edge(g;a;b),
lg-remove: lg-remove(g;n),
lg-in-edges: lg-in-edges(g;x),
member: t ∈ T,
top: Top,
subtype_rel: A ⊆r B,
labeled-graph: LabeledGraph(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
nat: ℕ,
lg-size: lg-size(g),
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
le: A ≤ B,
less_than: a < b,
less_than': less_than'(a;b),
false: False,
not: ¬A,
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),
bnot: ¬bb,
assert: ↑b,
int_iseg: {i...j},
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
cand: A c∧ B,
subtract: n - m,
spreadn: spread3,
pi2: snd(t),
pi1: fst(t)
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:
2016_05_17-AM-10_09_39
Last ObjectModification:
2016_01_18-AM-00_24_10
Theory : process-model
Home
Index