Nuprl Lemma : lg-search-property
∀[T:Type]. ∀G:LabeledGraph(T). ∀P:T ⟶ 𝔹. (↑isl(lg-search(G;x.P[x])) ⇐⇒ 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]),
labeled-graph: LabeledGraph(T),
assert: ↑b,
isl: isl(x),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
lg-exists: lg-exists(G;x.P[x]),
lg-search: lg-search(G;x.P[x]),
let: let,
lg-label2: lg-label2(g;x),
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
int_seg: {i..j-},
implies: P ⇒ Q,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
isl: isl(x),
assert: ↑b,
bfalse: ff,
false: False,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
lelt: i ≤ j < k,
true: True
Latex:
\mforall{}[T:Type]. \mforall{}G:LabeledGraph(T). \mforall{}P:T {}\mrightarrow{} \mBbbB{}. (\muparrow{}isl(lg-search(G;x.P[x])) \mLeftarrow{}{}\mRightarrow{} lg-exists(G;x.\muparrow{}P[x]))
Date html generated:
2016_05_17-AM-10_18_42
Last ObjectModification:
2016_01_18-AM-00_20_58
Theory : process-model
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