Nuprl Lemma : fpf-vals-singleton
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[P:A ⟶ 𝔹]. ∀[f:x:A fp-> B[x]]. ∀[a:A].
(fpf-vals(eq;P;f) = [<a, f(a)>] ∈ ((x:A × B[x]) List)) supposing ((∀b:A. (↑(P b) ⇐⇒ b = a ∈ A)) and (↑a ∈ dom(f)))
Proof
Definitions occuring in Statement :
fpf-vals: fpf-vals(eq;P;f),
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
cons: [a / b],
nil: [],
list: T List,
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
fpf-vals: fpf-vals(eq;P;f),
let: let,
fpf: a:A fp-> B[a],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
fpf-dom: x ∈ dom(f),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
not: ¬A,
false: False,
uiff: uiff(P;Q),
guard: {T},
or: P ∨ Q,
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b)
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:x:A fp-> B[x]]. \mforall{}[a:A].
(fpf-vals(eq;P;f) = [<a, f(a)>]) supposing ((\mforall{}b:A. (\muparrow{}(P b) \mLeftarrow{}{}\mRightarrow{} b = a)) and (\muparrow{}a \mmember{} dom(f)))
Date html generated:
2016_05_16-AM-11_19_00
Last ObjectModification:
2016_01_17-PM-03_50_24
Theory : event-ordering
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