Nuprl Lemma : member-fpf-vals
∀[A:Type]
∀eq:EqDecider(A)
∀[B:A ⟶ Type]
∀P:A ⟶ 𝔹. ∀f:x:A fp-> B[x]. ∀x:A. ∀v:B[x].
((<x, v> ∈ fpf-vals(eq;P;f)) ⇐⇒ {((↑x ∈ dom(f)) ∧ (↑(P x))) ∧ (v = f(x) ∈ B[x])})
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],
l_member: (x ∈ l),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: 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 :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
fpf: a:A fp-> B[a],
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf-vals: fpf-vals(eq;P;f),
pi1: fst(t),
pi2: snd(t),
let: let,
member: t ∈ T,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
subtype_rel: A ⊆r B,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
bnot: ¬bb,
cand: A c∧ B,
deq: EqDecider(T),
eqof: eqof(d),
true: True,
bor: p ∨bq
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{}x:A. \mforall{}v:B[x].
((<x, v> \mmember{} fpf-vals(eq;P;f)) \mLeftarrow{}{}\mRightarrow{} \{((\muparrow{}x \mmember{} dom(f)) \mwedge{} (\muparrow{}(P x))) \mwedge{} (v = f(x))\})
Date html generated:
2016_05_16-AM-11_18_19
Last ObjectModification:
2016_01_17-PM-03_52_54
Theory : event-ordering
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