Nuprl Lemma : accum-class-programmable
∀[Info,A,B:Type]. ∀[X:EClass(A)]. ∀[base:A ⟶ B]. ∀[f:B ⟶ A ⟶ B].
(accum-class(b,a.f[b;a];a.base[a];X)
= λB,r. if (#(B 0) =z 1)
then if (#(r) =z 1) then {f[only(r);only(B 0)]} else {base[only(B 0)]} fi
else {}
fi |λi.X,(self)'|
∈ EClass(B))
Proof
Definitions occuring in Statement :
rec-combined-class: f|X,(self)'|,
accum-class: accum-class(a,x.f[a; x];x.base[x];X),
eclass: EClass(A[eo; e]),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
bag-only: only(bs),
bag-size: #(bs),
single-bag: {x},
empty-bag: {}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
so_apply: x[s1;s2],
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
so_apply: x[s],
nequal: a ≠ b ∈ T ,
so_lambda: λ2x y.t[x; y],
eclass: EClass(A[eo; e]),
rec-combined-class: f|X,(self)'|,
in-eclass: e ∈b X,
eclass-val: X(e),
sv-class: Singlevalued(X),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eq_int: (i =z j),
accum-class: accum-class(a,x.f[a; x];x.base[x];X),
strongwellfounded: SWellFounded(R[x; y]),
ge: i ≥ j ,
es-E-interface: E(X),
sq_stable: SqStable(P),
es-prior-val: (X)',
nat_plus: ℕ+,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b]
Latex:
\mforall{}[Info,A,B:Type]. \mforall{}[X:EClass(A)]. \mforall{}[base:A {}\mrightarrow{} B]. \mforall{}[f:B {}\mrightarrow{} A {}\mrightarrow{} B].
(accum-class(b,a.f[b;a];a.base[a];X)
= \mlambda{}B,r. if (\#(B 0) =\msubz{} 1)
then if (\#(r) =\msubz{} 1) then \{f[only(r);only(B 0)]\} else \{base[only(B 0)]\} fi
else \{\}
fi |\mlambda{}i.X,(self)'|)
Date html generated:
2016_05_17-AM-08_12_54
Last ObjectModification:
2016_01_17-PM-02_51_27
Theory : event-ordering
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