Nuprl Lemma : lg-search-minimal
∀[T:Type]. ∀[G:LabeledGraph(T)]. ∀[P:T ⟶ 𝔹].
∀[n:ℕlg-size(G)]. outl(lg-search(G;x.P[x])) ≤ n supposing ↑P[lg-label(G;n)] supposing lg-exists(G;x.↑P[x])
Proof
Definitions occuring in Statement :
lg-search: lg-search(G;x.P[x]),
lg-exists: lg-exists(G;x.P[x]),
lg-label: lg-label(g;x),
lg-size: lg-size(g),
labeled-graph: LabeledGraph(T),
int_seg: {i..j-},
outl: outl(x),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
lg-exists: lg-exists(G;x.P[x]),
lg-label2: lg-label2(g;x),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
lg-search: lg-search(G;x.P[x]),
let: let,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
int_seg: {i..j-},
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
nat: ℕ,
ifthenelse: if b then t else f fi ,
outl: outl(x),
rev_implies: P ⇐ Q,
guard: {T},
exists: ∃x:A. B[x],
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
nequal: a ≠ b ∈ T ,
less_than: a < b,
squash: ↓T,
isl: isl(x),
true: True,
le: A ≤ B,
so_lambda: λ2x.t[x]
Latex:
\mforall{}[T:Type]. \mforall{}[G:LabeledGraph(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}].
\mforall{}[n:\mBbbN{}lg-size(G)]. outl(lg-search(G;x.P[x])) \mleq{} n supposing \muparrow{}P[lg-label(G;n)]
supposing lg-exists(G;x.\muparrow{}P[x])
Date html generated:
2016_05_17-AM-10_18_47
Last ObjectModification:
2016_01_18-AM-00_21_03
Theory : process-model
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