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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
lg-connected: lg-connected(g;a;b),
rel_plus: R+,
infix_ap: x f y,
exists: ∃x:A. B[x],
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
lelt: i ≤ j < k,
nat_plus: ℕ+,
guard: {T},
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
rel_exp: R^n,
eq_int: (i =z j),
cand: A c∧ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True
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:
2016_05_17-AM-10_10_14
Last ObjectModification:
2016_01_18-AM-00_24_36
Theory : process-model
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