Nuprl Lemma : filter-fpf-vals
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[P,Q:A ⟶ 𝔹]. ∀[f:x:A fp-> B[x]].
(filter(λpL.Q[fst(pL)];fpf-vals(eq;P;f)) ~ fpf-vals(eq;λa.((P a) ∧b (Q a));f))
Proof
Definitions occuring in Statement :
fpf-vals: fpf-vals(eq;P;f),
fpf: a:A fp-> B[a],
filter: filter(P;l),
deq: EqDecider(T),
band: p ∧b q,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
fpf-vals: fpf-vals(eq;P;f),
fpf: a:A fp-> B[a],
let: let,
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[P,Q:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:x:A fp-> B[x]].
(filter(\mlambda{}pL.Q[fst(pL)];fpf-vals(eq;P;f)) \msim{} fpf-vals(eq;\mlambda{}a.((P a) \mwedge{}\msubb{} (Q a));f))
Date html generated:
2016_05_16-AM-11_18_33
Last ObjectModification:
2016_01_17-PM-03_50_37
Theory : event-ordering
Home
Index